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Abbring, Jaap H.; Campbell, Jeffrey R.; Yang, Nan

Working Paper Simple Markov-perfect industry dynamics

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Suggested Citation: Abbring, Jaap H.; Campbell, Jeffrey R.; Yang, Nan (2010) : Simple Markov- perfect industry dynamics, Working Paper, No. 2010-21, Federal Reserve Bank of Chicago, Chicago, IL

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Simple Markov-Perfect Industry Dynamics

Jaap H. Abbring, Jeffrey R. Campbell, and Nan Yang

November 30, 2010

Federal Reserve Bank of Chicago WP 2010-21

Simple Markov-Perfect Industry Dynamics∗

Jaap H. Abbring† Jeﬀrey R. Campbell‡ Nan Yang§

November 30, 2010

Abstract

This paper develops a tractable model for the computational and empirical analysis of infinite-horizon oligopoly dynamics. It features aggregate demand uncertainty, sunk entry costs, stochastic idiosyncratic technological progress, and irreversible exit. We develop an algorithm for computing a symmetric Markov-perfect equilibrium quickly by finding the fixed points to a finite sequence of low-dimensional contraction mappings. If at most two heterogenous firms serve the industry, the resuilt is the unique “natural” equilibrium in which a high profitability firm never exits leaving behind a low profitability competitor. With more than two firms, the algorithm always finds a natural equilibrium. We present a simple rule for checking ex post whether the calculated equilibrium is unique, and we illustrate the model’s application by assessing how price collusion impacts consumer and total surplus in a market for a new product that requires costly development. The results confirm Fershtman and Pakes’(2000) finding that collusive pricing can increase consumer surplus by stimulating product development. A distinguishing feature of our analysis is that we are able to assess the results’ robustness across hundreds of parameter values in only a few minutes on an off-the-shelf laptop computer.

∗We thank R. Andrew Butters and Xiye Yang for their expert research assistance. †CentER, Department of Econometrics & OR, Tilburg University. E-mail: [email protected] ‡Federal Reserve Bank of Chicago. E-mail: [email protected] §VU University Amsterdam and Tinbergen Institute. E-mail: [email protected] JEL Code: L13 Keywords: Sunk costs, Demand uncertainty, Markov-perfect equilibrium, Learning-by-doing, Technology innovation 1 Introduction

This paper supplies fast, effective, and simple computational methods for important special cases of Ericson and Pakes’(1995) model of dynamic oligopoly. These cases feature aggregate uncertainty, sunk entry costs, and stochastic firm-specific technological progress; but they exclude investment decisions other than entry and exit. This simplification facilitates a range of equilibrium characterization, existence, and uniqueness results that are not available for the more general framework. Moreover, it enables the development of algorithms that calculate equilibria by finding the fixed points of a finite sequence of low-dimensional contraction mappings. These results can be used to explore some key aspects of Ericson and Pakes’ model with very low computational cost. This is often useful in itself, and can serve as a first stage of a richer analysis with a more complex specification. Substantial methodological progress in the computation of Markov-perfect equilibria followed Ericson and Pakes’ original presentation of their framework. Nevertheless, Doraszelski and Pakes(2007) note that these methodological developments are only in their infancy and applications remain rare. This paper contributes to this literature by developing rela- tively rich analytical results and effective computational methods for a comparatively simple model. It shares this approach with Abbring and Campbell’s (2010) analysis of last-in first- out oligopoly dynamics. They consider a dynamic extension of Bresnahan and Reiss’(1990) static entry model that can naturally be applied to the empirical analysis of market level entry and exit data (Abbring, Campbell, and Yang, 2010). Timing and expectational assumptions simplify its equilibrium analysis: Otherwise homogeneous firms move sequentially, oldest first; and older firms never exit expecting to leave a younger firm behind. The present paper contributes more directly to the analysis of Ericson and Pakes’ framework and its potential applications, because it allows for idiosyncratic technological progress in a model with simultaneously moving incumbent firms. Our results leverage one key insight into the structure of payoffs in a symmetric Markov- perfect equilibrium: If any firm chooses to exit with positive probability, then all identically situated firms must have an expected continuation value of zero. This allows us to calculate firms’ expected continuation values at some nodes of the game tree without knowing everything about how the game will proceed thereafter. Our results demonstrate how to use these initial calculations to recover all equilibrium payoffs and actions. For this task, it is very helpful to know beforehand that adding an active firm to an industry weakly reduces all other firms’ continuation values. We prove that this intuitive property must hold if at most two firms can serve the industry at one time. For the more general oligopoly case, we show that if a Markov-perfect equilibrium with such monotonicity exists, then it is essentially unique. In this case, the algorithm we propose always computes it. If no such equilibrium

1 exists, then our algorithm can be easily adapted to find all equilibria satisfying a desirable property we call “one-shot renegotiation proofness”. The remainder of this paper proceeds as follows. The next section presents the model’s primitives. It also discusses the equilibrium concept used, natural Markov-perfect equilibrium. As in Cabral(1993), the restriction to “natural” equilibrium requires no firm with high flow profits to exit leaving a lower-profitability rival in the market. Section3 covers the special case of a market that can support at most two active firms. The proofs of equilibrium existence and uniqueness are constructive, and so they naturally generate an algorithm for equilibrium computation. Its central steps find the fixed points of a finite number of low-dimensional contraction mappings. We apply the results to a numerical analysis of the effects of relaxing short-term price competition on welfare-enhancing product development, earlier explored by Fershtman and Pakes(2000). Section4 begins with extending the algorithm from duopoly model to accommodate three or more potentially heterogeneous firms. We then show that if a natural equilibrium in which adding incumbent firms weakly lowers continuation values exists, then it is essentially unique and the algorithm computes it. Next, we illustrate with an example that it is possible for entry to increase an incumbent’s expected discounted payoff. This counterintuitive effect of entry arises from the entry deterring effects of competition. Our analysis identifies two sources of equilibrium multiplicity, both of which require entry to raise an incumbent’s equilibrium payoff at some point. One arises from the failure of incumbent firms to coordinate on survival when this is mutually beneficial. We propose to exclude such coordination failures by requiring equilibria to be “one-shot renegotiation proof”. The other occurs when multiple mixed strategies leave incumbents indifferent between exit and continuation.

2 The Model

In Ericson and Pakes(1995), a countable number of firms with heterogeneous productivity levels serve a single industry. Entry requires the payment of a sunk cost, and exit allows firms to avoid per-period fixed costs of production. Surviving incumbent firms choose investments that stochastically improve their technologies. Exogenous stochastic increases in the knowledge stock outside the industry increase the quality of an outside good and, this way, decrease all incumbent firms’ profits simultaneously. These outside knowledge shocks are embodied in potential entrants to the industry, and therefore do not affect their profits. Two main changes to Ericson and Pakes’ primitive assumptions facilitate our demonstra- tion of Markov-perfect equilibrium uniqueness and our algorithm for its rapid computation. First, we assume that productivity evolves exogenously, instead of allowing firms to make costly investments in accelerating technological progress. Second, we replace the common

2 Start with Firm f gets f (Ct,Nt) π(Nt,Ct,Kt ) if active

Entry decision (t, 1)

a = 1; Pay ϕ & enter a = 0; Pass & earn 0

Entry a ∈ [0, 1] decision (t, 2)

Entry decision

(t, Jt + 1)

Simultaneous Nature chooses Kf using t+1 Go to next period continuation Π if ﬁrm f is active and with (Ct+1,Nt+1) decisions chooses Ct+1 ∼ Q(·|Ct).

Figure 1: The Sequence of Events and Actions within a Period negative shocks to the incumbents’ profits by general aggregate demand shocks that equally affect the profits of incumbent firms and potential entrants.

2.1 Primitive Assumptions

The model consists of a single oligopolistic market in discrete time t ∈ Z? ≡ {0, 1,...}.A countable number of firms potentially serve the market. These are indexed by f ∈ Z? × N. Below we refer to f as the firm’s name. At a given time t, some of the firms are active, and the remaining producers are inactive. Each active firm f has an idiosyncratic productivity f ˇ type Kt that takes values in K ≡ {1,..., k}. Stack the numbers of active firms with each ˇ productivity level at time t into the k × 1 vector Nt, the market structure. Initially, no firms serve the market: N0 equals a vector of zeros. Subsequently; entry, stochastic productivity improvement, and exit determine its evolution.

3 Figure1 illustrates the sequence of events and actions within a period t. It begins with the inherited values of two state variables, Nt and a scalar index of demand Ct ∈ [ˆc, cˇ], with cˇ < ∞. With these in place, the active participants receive their profits from serving the 1 market. For a type Kt firm facing the market structure Nt, these equal π(Nt,Ct,Kt). We assume that a firm’s flow profit decreases with the number and productivity of its competitors and increases with its own productivity. For this assumption’s formal statement, ˇ we use ιk to denote a k × 1 vector with a one in its kth position and zeros elsewhere, and set

ι0 ≡ 0. This allows us to denote a market structure with at least one type k ﬁrm with n + ιk.

Assumption 1 (Monotone Producer Surplus). For all productivity types k ∈ K, demand kˇ states c ∈ [ˆc, cˇ], and market structures n ∈ Z?:

1. π(ιk + n, c, k) ≤ πˇ < ∞ for all c ∈ [ˆc, cˇ]

2. π(ιk + n, c, k) ≤ π(ιk + n, d, k) for all c, d ∈ [ˆc, cˇ] such that c < d.

3. π(ιk + n + ιl, c, k) < π (ιk + n + ιl−1, c, k) for all l ∈ K;

4. π(ιk + n, c, k) → −κ(k) < 0 as the number of ﬁrms in n goes to inﬁnity; and

5. π(ιk + ιl + n, c, k) ≤ π(ιk + ιl + n, c, l) for all k, l ∈ K such that k < l, with strict inequality hold for some c ∈ [ˆc, cˇ].

In item4, κ(k) represents a type k firm’s per-period fixed cost. After production, firms with names in {t} × N make entry decisions sequentially in the order of their names, starting with (t, 1). These continue until a firm chooses to remain out of the industry. We denote the number of entrants in period t with Jt, so the name of the first potential entrant choosing to stay out of the market and thereby ending its entry stage is (t, Jt + 1). The cost of entry is ϕ > 0. After paying this cost, the entrant immediately joins the set of active firms with productivity type 1.2 A firm with an entry opportunity cannot delay its choice, so the payoff to staying out of the industry is zero. After the entry decisions, all active firms— including those that just entered the market— decide simultaneously between survival and exit. Exit is irreversible but otherwise costless. It allows firms to avoid future periods’ fixed production costs. Firms’ entry and exit decisions maximize their expected profit streams discounted with a factor β < 1.

In the period’s final stage, Ct and the firms’ productivity types evolve. The demand index evolves exogenously according to a nonnegative first-order Markov process bounded between

1 We leave this undefined if the Kt’th element of Nt equals zero. 2Since entrants’ productivity types evolve before their first period of production, we can use the distri- f f bution of Kt+1 given Kt = 1 to distribute new firms’ types nontrivially. That is, the assumption that all f entrants have Kt = 1 is not overly restrictive.

4 cˆ andc ˇ. We denote the conditional distribution of Ct+1 with Q (c Ct) ≡ Pr (Ct+1 ≤ c Ct), and the corresponding probability density function with q(· Ct). Each ﬁrm’s idiosyncratic productivity type follows an independent Markov chain with a common (kˇ × kˇ) transition f 0 f matrix Π. Its typical element is Πk,k0 ≡ Pr Kt+1 = k Kt = k . Following Ericson and Pakes(1995), we assume that the idiosyncratic productivity types never regress:

Assumption 2 (No Productivity Regress). Π is upper diagonal.

f f We further assume that Kt+1 (weakly) stochastically increases with Kt .

Assumption 3 (Monotone Productivity Dynamics). For all k0, k, l ∈ K such that k < l,

f 0 f f 0 f Pr Kt+1 ≥ k Kt = k ≤ Pr Kt+1 ≥ k Kt = l .

This assumption gives high technology ﬁrms no worse advancement opportunities than low technology ﬁrms have.

2.2 Markov-Perfect Equilibrium

A Markov-perfect equilibrium is a subgame-perfect equilibrium in strategies that are only contingent on payoﬀ-relevant variables. For a potential participant f = (t, j) contemplating f entry these are Ct and the market structure Mt just after f’s possible entry. The latter is f period t’s initial market structure Nt augmented with j type 1 entrants: Mt ≡ Nt + jι1.

Denote the market structure after the period’s final entry with ME,t ≡ Nt + ι1Jt. If firm f is contemplating survival in period t, the payoff-relevant variables are this market structure, f the current demand index (Ct), and its productivity type (Kt ). f f A Markov strategy for firm f is a pair (aE, aS) of functions

f kˇ f kˇ aE : Z? × [ˆc, cˇ] −→ [0, 1] and aS : Z? × [ˆc, cˇ] × K −→ [0, 1].

f This strategy’s entry rule aE assigns a probability of becoming active given an entry oppor- f f tunity to each possible value of (Mt ,Ct). Similarly, its exit rule aS assigns a probability of being active in the next period given that the firm is currently active to each possible value f of its payoff-relevant state (ME,t,Ct,Kt ). Since calendar time is not payoff-relevant, we hereafter drop the t subscript from all variables. A symmetric equilibrium is an equilibrium in which all firms follow the same strategy (aE, aS). In the remainder of the paper, we focus on symmetric equilibria and drop the superscript f from the firms’ common strategy. Throughout the paper, we will focus on equilibria in which a high productivity firm never exits when a low productivity competitor survives. Such equilibria are natural, because a high productivity firm earns strictly higher flow profit in each period than a low productivity firm. Formally, we define a natural Markov-perfect equilibrium as follows:

5 Definition 1. A natural Markov-perfect equilibrium is a symmetric Markov-perfect equilibrium in a strategy (aE, aS) such that for all k, l ∈ K such that k < l; mk ≥ 1, ml ≥ 1, and aS(m, c, k) > 0 together imply that aS(m, c, l) = 1. Cabral(1993) restricts attention to similar natural equilibria in a model with deterministic productivity progression. Firms’ expected discounted profits at each node of the game depend on that node’s payoff-relevant state variables when they all use Markov strategies. The payoffs in two of each period’s nodes are of particular interest, the post-entry value and the post-survival value.

The post-entry value vE(ME,C,K) equals the expected discounted profits of a type K firm in a market with demand index C and market structure ME just after all entry decisions have been sequentially realized. Since it gives the payoffs to a potential producer from entering in each possible market structure that could arise from other players subsequent entry decisions, it determines optimal entry choices. The post-survival value vS(MS,C,K) equals the expected discounted profits of a type K firm facing demand index C and market structure MS just after all survival decisions have been realized. It gives the payoffs to a surviving firm in each possible market structure following firms’ simultaneous continuation decisions, so it is central to the analysis of exit.

The value functions vE and vS satisfy

vE(mE, c, k) = aS(mE, c, k)E [vS(MS, c, k) ME = mE] (1) and

0 0 0 0 0 0 vS(mS, c, k) = βE [π(N ,C ,K ) + vE(ME,C ,K ) MS = mS,C = c, K = k] . (2) Here and throughout, we denote the variable corresponding to X in the next period with X0. The conditional expectation in (1) is computed given that the firm of interest continues, and embodies the use of aS by all other active firms. In fact, the only nontrivial randomness it embodies arises from firms’ possible use of mixed strategies. The conditional expectation in

(2) accounts for the use of aE by all potential participants with entry opportunities in the next period as well as the exogenous evolutions of C and the ﬁrms’ productivity types.3

For (aE, aS) to form a symmetric Markov-perfect equilibrium, it is necessary and suﬃcient that no ﬁrm can gain from a one-shot deviation from (aE, aS) (e.g. Fudenberg and Tirole, 1991, Theorem 4.2):

aE(m, c) ∈ arg max a {E [vE (ME, c, 1) M = m] − ϕ} and (3) a∈[0,1]

aS(mE, c, k) ∈ arg max a E [vS(MS, c, k) ME = mE] . (4) a∈[0,1]

3Section ?? of this paper’s online appendix presents the two conditional distributions underlying the conditional expectations in (1) and (2) in detail.

6 The conditional expectations in (3) and (4) are computed like those in (1) and (2). For example, E [vE (ME, c, 1) M = m] is the payoff, gross of the entry cost ϕ, that a potential participant in state (m, c) expects from entering if all firms with entry opportunities later in the period use the entry rule aE and the value of ending up as a type 1 firm in a market with structure mE and c consumers equals vE (mE, c, 1).

Together, conditions (1)–(4) are necessary and sufficient for a strategy (aE, aS) to form a symmetric Markov-perfect equilibrium with payoffs vE and vS. Before proceeding to examine the set of natural Markov-perfect equilibria, consider one uninteresting source of equilibrium multiplicity. With an equilibrium in hand, change one player’s action at a particular node of the game. If this change gives the same payoff to the player in question and all other player’s equilibrium actions at that node remain optimal, then this change forms a second equilibrium. In our model, this situation can arise when the payoff to entry equals zero and when the payoff to survival as the only firm of your type equals zero. To eliminate this difficulty, we require firms in such a situation to choose inactivity.

Deﬁnition 2. A Markov strategy (aS, aE) with corresponding payoﬀ vE defaults to inactivity if

• aS(m − mk × ιk + ιk, c, k) = 0 if vS(m, c, k) = 0

• aE(m, c) = 0 if vE(m, c, 1) = ϕ, for all k ∈ K and all c.

The remainder of the paper restricts attention to equilibria with strategies that default to inactivity, unless otherwise mentioned.4

3 Duopoly

It is helpful to begin the model’s analysis with one additional restriction: At most two ﬁrms can be active at once. Throughout this section, we represent duopoly market structures with

ιk + ιl with k, l ∈ K ∪ {0}. The following lemma arises from this simpliﬁcation.

Lemma 1 (Monotone Payoﬀs in the Heterogenous-Duopoly Model). In a natural Markov- perfect equilibrium, for all c ∈ [ˆc, cˇ] and k ∈ K, vE(2ιk, c, k) ≤ vE(ιk, c, k) and vS(2ιk, c, k) ≤ vS(ιk, c, k).

Proof. See AppendixA.

4Note that we do not restrict the game’s strategy space to include only strategies that default to inactivity.

7 Figure 2: Reduced-form Representation of the Duopoly Continuation Game

Survive Exit

vS(2, c) v (1, c) Survive S vS(2, c) 0 0 0 Exit vS(1, c) 0

Note: In each cell, the upper-left expression gives the row player’s payoﬀ. Please see the text for further details.

Lemma1 states that a duopolist facing a rival of the same type always has a lower value than it would have without the rival present. With its help, we develop the duopoly model’s analysis in three stages. First, we examine the special case without heterogeneity, kˇ = 1. This introduces the model’s most important moving parts without undue complication. We then generalize this slightly in Section 3.2 by adding a second productivity type and walking through the procedure for equilibrium calculation. Section Section 3.3 formalizes this procedure into an algorithm and then establishes equilibrium existence and uniqueness results. Finally, Section 3.4 uses this algorithm for numerical analysis of the eﬀects of technological progress and demand uncertainty on industry dynamics. This illustration demonstrates that the natural Markov-perfect equilibrium of this model can be easily computed.

3.1 One Productivity Type

When ﬁrms have identical productivity types by assumption, the restriction to a natural equilibrium merely requires symmetry of players’ strategies. Here, we construct a symmetric Markov-perfect equilibrium for this case in three steps. The type distribution is trivial, so we write π(N,C, 1) as π(N,C) and make the analogous substitution for the value functions throughout this example’s development.

Step 1: Calculation of vE(2, ·), vS(2, ·), and aE(2, ·) The equilibrium construction begins with a characterization of the duopoly payoffs vE(2, ·) and vS(2, ·). In a Markov-perfect equilibrium, the survival rule aS(2, c) satisfies (4): Given c, it is a Nash equilibrium of the static simultaneous-move game with payoffs given by the expected continuation values. Figure2 gives the reduced-form representation of this game with the two pure strategies “Survive” and “Exit”. The upper-left expression in each cell is the row player’s payoff. Both

firms receive the duopoly post-survival payoff vS(2, c) following joint survival. This payoff

8 adds the discounted duopoly flow payoff π(2,C0) to the the discounted duopoly post-entry 0 payoff vE(2,C ). Consequently, it satisfies a special case of Equation (2):

0 0 vS(2, c) = βE [π(2,C ) + vE(2,C ) C = c] .

A ﬁrm that survives while its rival exits earns the monopoly post-survival value vS(1, c).

Suppose that vS(2, c) > 0. Lemma1 guarantees that vS(1,C) > 0, so in this case

“Survive” is a dominant strategy. If instead vS(2, c) < 0, then a symmetric equilibrium strategy must put positive probability on “Exit”. That pure strategy’s payoﬀ always equals zero. Since vE(2, c) equals the symmetric equilibrium payoﬀ to this game, these facts together yield the following special case of Equation (1):

vE(2, c) = max {0, vS(2, c)} (5) n 0 0 o = max 0, βE [π(2,C ) + vE(2,C ) C = c] .

The right-hand side defines a contraction mapping, so this necessary condition uniquely determines vE(2, ·) and, using (2), vS(2, ·). This is the key technical insight that makes the calculation of the model’s Markov-perfect industry dynamics simple. Although duopoly is not an absorbing state for the industry, we can calculate the equilibrium duopoly payoffs without knowledge of the firms’ payoffs in possible future market structures. This is because firms’ common post-entry value in a symmetric equilibrium equals zero unless joint continuation is individually profitable. With the duopoly post-entry value in hand, we can proceed to the problem of a potential entrant facing a single incumbent. By Equation (3), this firm enters if vE(2, c) > ϕ and stays out of the market if vE(2, c) ≤ ϕ. For all c,

aE(2, c) = I {vE(2, c) > ϕ} .

Note that strategy defaults to inactivity. When C has an atomless distribution, this strategy almost surely prescribes the same action as any other entry strategy consistent with proﬁt maximization that does not default to inactivity. For this reason, our requirement that the potential entrant default to inactivity has no substantial economic content.

Step 2: Calculation of vE(1, ·), vS(1, ·), aE(1, ·), and aS(1, ·) We proceed to consider the monopoly payoﬀs, a potential entrant’s decision to enter an empty market, and an incumbent monopolist’s survival decision. Because an incumbent monopolist choosing to survive will earn vS(1, c), the post-entry value to a monopolist in (1) reduces to

vE(1, c) = max {0, vS(1, c)} n h io 0 0 0 0 0 = max 0, βE π(1,C ) + aE(2,C )vE(2,C ) + (1 − aE(2,C )) vE(1,C ) C = c .

9 Given vE(2, ·) and aE(2, ·) from Step 1, the right-hand side deﬁnes a contraction mapping that uniquely determines vE(1, ·) and, using Equation (2), vS(1, ·). It is not diﬃcult to demonstrate that the vE(1, c) and vS(1, c) so constructed always weakly exceed, respectively, vE(2, c) and vS(2, c) from Step 1; so that the constructed value functions are consistent with the requirements of Lemma1. Just as with a potential duopolist, we select the unique entry rule for a potential monopolist that defaults to inactivity. Since vE(2, c) ≤ vE(1, c), this is

aE(1, c) = I {vE(1, c) > ϕ} .

By (4), a monopolist chooses survival in demand states c such that vS(1, c) > 0 and exit if vS(1, c) < 0. Our equilibrium construction uses the unique monopoly survival rule that defaults to inactivity:

aS(1, c) = I {vS(1, c) > 0} .

Step 3: Calculate aS(2, ·) The ﬁrst two steps have determined the only possible post- entry and post-survival values, as well as an entry rule and a monopoly survival rule that are consistent with them. This last step completes the equilibrium strategy’s construction by determining a duopoly survival rule that satisﬁes (4).

As we noted above in Step 1, equilibrium requires aS(2, c) = 1 if vS(2, c) > 0. All that remains undetermined is the survival rule when vS(2, c) ≤ 0. If profit maximization would require even a monopolist to exit (i.e. vS(1, c) ≤ 0), then both duopolists exit for sure and aS(2, c) = 0. If instead vS(1, c) > 0, then the reduced-form continuation game above has no pure strategy equilibrium. In its unique mixed-strategy equilibrium, each firm’s survival probability leaves its rival indifferent between exiting (and getting a payoff of zero for sure) and surviving. That is, in demand states c such that vS(2, c) ≤ 0 and vS(1, c) > 0, the indifference condition

aS(2, c)vS(2, c) + (1 − aS(2, c)) vS(1, c) = 0

5 uniquely determines aS(2, c).

Illustration of the Constructed Equilibrium The entry and survival rules so calculated form our equilibrium. Figure3 plots the payoffs for a particular numerical example. In it π(c, n) = c$(n) − κ with κ > 0 and $(2) < $(1). We also choose the stochastic process for C so that its current value has no influence on the equilibrium probability of a firm entering

5 The mixed-strategy so derived prescribes that both ﬁrms exit for sure if vS(1, c) = 0, as the required by our restriction to strategies that default to inactivity.

10 Figure 3: Equilibrium Payoﬀs in the Homogeneous Duopoly Example

Expected Monopoly Payoﬀ vE(1, c)

Expected Duopoly Payoﬀ vE(2, c)

cˆ c1 c1 c2 c2 cˇ ( A ) ( B ) ( C ) or exiting in any future period. Specifically, C0 = c with probability 1 − λ and equals a draw from a uniform distribution over [ˆc, cˇ] with the complementary probability. This and the affine specification of π(c, n) together guarantee that vE(1, c) and vE(2, c) are piecewise linear in c.

The lower (continuous) function in grey gives the duopoly post-entry value, vE(2, c). By construction, this is identical to the expected discounted proﬁts of a duopolist facing a rival that will never exit ﬁrst. It equals zero for c ≤ c2. Thereafter it rises π(2)/(1 − β(1 − λ)) for each extra consumer. For c > c2, entry into a market with one incumbent is optimal.

The monopoly post-entry value vE(1, c) equals zero for demand levels c ≤ c1, and it increases π(1)/(1 − β(1 − λ)) with each extra consumer until reaching c2. . When c > c2, the period’s entry stage always ends with two firms, so the equilibrium payoff to a firm that began the period as a monopolist drops to the grey expected duopoly payoff. The disconnected line segment above this gives the expected payoff to a firm that finds itself as a monopolist after

11 the period’s entry stage is complete.6 Given this value function, the equilibrium strategy for a potential entrant facing an empty market is aE(1, c) = I {vE(1, c) > ϕ} ≡ I {c > c1}, and the analogous continuation rule for an incumbent monopolist is aS(1, c) = I {vE(1, c) > 0} =

I {vS(1, c) > 0} ≡ I {c > c1}. Duopolists’ common continuation strategy corresponds to the unique Nash equilibrium of the game in Figure2. Exit is a dominant strategy when c ∈ A, and survival is dominant when c ∈ C. When c ∈ B the ﬁrms mix over survival and exit.

3.2 Two Productivity Types

We now proceed to adapt the basic ideas presented above to the case with productivity heterogeneity. In the interest of expositional clarity, we denote the higher productivity type with the intuitive H (instead of 2) and the lower type with L (instead of 1). We construct this case’s unique natural Markov-perfect equilibrium in six steps. Just as before, these steps take us through a finite partition of the state space. In each of the first five steps, we compute the equilibrium payoffs in the states considered by finding the unique fixed point of a contraction mapping. The results from the completed steps are used as inputs in the following steps. Figure4 illustrates this sequence of computations. The construction ends by specifying the unique strategy that supports the equilibrium payoffs in the sixth step.7

Step 1: Calculation of vE(2ιH, ·, H) and vS(2ιH, ·, H) As depicted by the upper-left panel in Figure4, we start the equilibrium construction by considering a market populated by two type H firms. The analysis in this step is a carbon copy of the first step of the previous example. The simultaneous-move survival game between two type H firms is analogous to the one in Figure2, and Lemma1 guarantees that “Survive” is the dominant strategy if joint continuation gives both firms positive payoffs. Therefore, finding the fixed point of a contraction mapping analogous to that in (1) yields vE(2ιH, ·, H). The continuation payoff vS(2ιH, ·, H) immediately follows.

Step 2: Calculation of vE(ιL+ιH, ·, L), vS(ιL+ιH, ·, L), aE(ιL+ιH, ·), and aS(ιL+ιH, ·, L).

A type L ﬁrm that chooses to survive advances to H with probability ΠLH and remains unchanged with probability ΠLL ≡ 1 − ΠLH. In a natural MPE, the survival of the type L

ﬁrm guarantees survival of any type H rival, so the continuation value vE(ιL +ιH,C, L) must

6This would require some potential entrant to deviate from the equilibrium strategy. 7The Atari 400 computer appearing in the illustration went on sale in November 1979 and had 8K of RAM. It has more than enough computing power to calculate the unique equilibrium.

12 Figure 4: Equilibrium Computation for a Duopoly with Two Productivity Types

Number of type L ﬁrms Market structure with one ﬁrm of each type

0 1 0 2 1 0 1 0 2 1 START 2 1 1 0 0 2 1 1 0 0

Value functions calculated Number of type H ﬁrms

0 1 0 2 1 0 1 0 2 1 2 1 1 0 0 2 1 1 0 0

Value functions in hand

0 1 0 2 1 0 1 0 2 1 STOP 2 1 1 0 0 2 1 1 0 0

Note: There are ﬁve possible duopoly market structures. Each divided rectangle represents one of them, and each collection of ﬁve rectangles displays the value functions being calculated (in red) and the value functions already in hand (in blue) at one stage of the algorithm (which is Section 3.3’s Algorithm1 with kˇ = 2). satisfy n o vE(ιL + ιH, c, L) = max 0, vS(ιL + ιH, c, L)

n 0 0 =β max 0, ΠLLE [π(ιL + ιH,C , L) + vE(ιL + ιH,C , L)|C = c]

0 0 o + ΠLHE [π(2ιH,C , H) + vE(2ιH,C , H)|C = c] .

Since vE(2ιH, ·, ιH) is in hand from Step 1, this deﬁnes a contraction mapping in the desired value function. With its ﬁxed-point in hand, we can then easily compute vS(ιL +ιH, ·, L) and

aE(ιL + ιH, c) = I{vE(ιL + ιH, c, L) > ϕ},

aS(ιL + ιH, c, L) = I{vS(ιL + ιH, c, L) > 0}.

Step 3: Calculation of vE(ιH, ·, H), vS(ιH, ·, H), aS(ιH, ·, H), vE(ιL + ιH, ·, H), vS(ιL +

ιH, ·, H), and aS(ιL +ιH, ·, H). A market with a monopolist incumbent with type H attracts 0 an entrant next period if and only if aE(ιL + ιH,C ) = 1, so vE(ιH, ·, H) and vE(ιH + ιL, ·, H)

13 together satisfy n o vE(ιH, c, H) = max 0, vS(ιH, c, H)

n 0 0 0 =β max 0, E[π(ιH,C , H) + aE(ιL + ιH,C )vE(ιL + ιH,C , H) (6)

0 0 o + {1 − aE(ιL + ιH,C )} vE(ιH,C , H)|C = c] .

Step 2 determined aE(ιL+ιH, ·), so the only unknowns in (6) are the value functions. Since a type H duopolist facing a type L rival becomes a monopolist if and only if aS(ιL +ιH, ·, L) = 0, these value functions must also satisfy

vE(ιL + ιH, c, ιH)

= aS(ιL + ιH, c, L)vS(ιL + ιH, c, ιH) + {1 − aS(ιL + ιH, c, L)} vE(ιH, c, H)

n 0 0 = aS(ιL + ιH, c, L)β ΠLLE[π(ιH + ιL,C , H) + vE(ιL + ιH,C , H)|C = c] (7)

0 0 o + ΠLHE[π(2ιH,C , H) + vE(2ιH,C , H)|C = c]

+ {1 − aS(ιL + ιH, c, L)} vE(ιH, c, H).

We have vE(2ιH, ·, H) from Step 1 and aS(ιL + ιH, ·, L) from Step 2, so together, (6) and

(7) determine vE(ιH, ·, H) and vE(ιL + ιH, ·, H). Obtaining vS(ιH, ·, H) and vS(ιL + ιH, ·, H) from these is straightforward. Since we seek a natural equilibrium, the survival strategies of interest must satisfy

aS(ιH, c, H) = aS(ιL + ιH, c, H) = I{vS(ιH, c, H) > 0}.

Step 4: Calculation of vE(2ιL, ·, L), aE(2ιL, ·), and vS(2ιL, ·, L). Next, we consider a duopoly market with two type L firms. If both firms choose survival, then their idiosyncratic shocks could change the market structure to either of the duopoly structures considered in Steps 1-3 or leave it unchanged. Lemma1 guarantees that if the value of simultaneous survival to either incumbent is positive, then joint continuation is the only Nash equilibrium outcome of their survival game. Therefore, vE(2ιL, ·, L) satisfies n o vE(2ιL, c, L) = max 0, vS(2ιL, c, L)

n 2 0 0 =β max 0, ΠLLE [π(2ιL,C , L) + vE(2ιL,C , L)|C = c] 0 0 + ΠLLΠLHE [π(ιL + ιH,C , L) + vE(ιL + ιH,C , L)|C = c] 0 0 + ΠLHΠLLE [π(ιL + ιH,C , H) + vE(ιL + ιH,C , H)|C = c] 2 0 0 o + ΠLHE [π(2ιH,C , H) + vE(2ιH,C , H)|C = c] .

14 The only unknown on its righthand side is vE(2ιL, ·, L), so we can use this Bellman equation to calculate it. With this in hand, we construct the rule for entry into a market with one type L incumbent as

aE(2ιL, c) = I{vE(2ιL, c, L) > ϕ}.

Moreover, it is straightforward to determine vS(2ιL, ·, L).

Step 5: Calculation of vE(ιL, ·, L), vS(ιL, ·, L), aE(ιL, ·), and aS(ιL, ·, L). If a type L monopolist chooses survival, then one of four market structures will prevail in the next period, depending on the incumbent’s idiosyncratic shock and on the decision of a potential entrant:

n o vE(ιL, c, L) = max 0, vS(ιL, c, L)

n h 0 0 0 = max 0, ΠLLE π(ιL,C , L) + aE(2ιL,C )vE(2ιL,C , L)

0 0 i + {1 − aE(2ιL,C )} vE(ιL,C , L)|C = c (8)

h 0 0 0 + ΠLHE π(ιH,C , H) + aE(ιL + ιH,C )vE(ιL + ιH,C , H)

0 0 io + {1 − aE(ιL + ιH,C )} vE(ιH,C , H)|C = c .

Given the quantities calculated in Steps 1–4, the righthand side of (8) deﬁnes a contraction mapping with vE(ιL, ·, L) as its ﬁxed point. With this, it is straightforward to compute vS(ιL, ·, L), which gives the survival rule

aS(ιL, c, L) = I{vS(ιL, c, L) > 0}.

Since vE(2ιL, c, L) ≤ vE(ιL, c, L), the entry rule for a potential monopolist can be written as

aE(ιL, c) = I {vE(ιL, c) > ϕ}

Step 6: Calculation of aS(2ιH, ·, H) and aS(2ιL, ·, L). Steps 1–5 have determined all equilibrium continuation values, entry strategies, and survival strategies for firms facing no identical rival. All that remains is to determine the exit strategies for duopolies of identical firms. Their construction parallels that from the case with homogeneous firms: Unless either survival or exit is a dominant strategy, both firms mix between the two pure actions to leave

15 each other indiﬀerent between them.  1 if v (2ι , c, L) > 0,  S L vS (ιL,c,L) aS(2ιL, c, L) = if vS(2ιL, c, L) ≤ 0 and vS(ιL, c, L) > 0 vS (ιL,c,L)−vS (2ιL,c,L)  0 otherwise.  1 if v (2ι , c, H) > 0,  S H vS (ιH,c,H) aS(2ιH, c, H) = if vS(2ιH, c, H) ≤ 0 and vS(ιH, c, H) > 0 vS (ιH,c,H)−vS (2ιH,c,H)  0 otherwise.

This concludes the equilibrium construction. Although adding productivity heterogeneity increased the number of steps required, calculating the ﬁxed point of a contraction mapping on a low-dimensional function space remains the most computationally intensive required task.

3.3 Equilibrium Existence, Uniqueness, and Computation

We next extend the six-step calculation of duopoly equilibrium with kˇ = 2 to allow for arbitrary kˇ.The resulting algorithm consists of two procedures, which we present as flow charts in Procedures1 and2. The first computes all payoffs, survival strategies for duopolists facing strictly higher productivity types, and strategy for a potential entrant facing an incumbent. The second procedure calculates the survival strategies for duopoly incumbents with weakly higher productivity types and the strategy for a potential entrant facing an empty market. In Procedure1’s flow chart, h indexes the productivity type for the weakly better firm, and l for the weakly worse firm. In the course of its execution, h decreases from kˇ to 1. For each level of h, l decreases from h to 1. For any pair of (h, l); the post-entry value wE to the type l firm that faces a type h rival is computed as the fixed point of Th,l. This functional operator is defined by the recursive condition for wE(ιl + ιh, ·, l). The type l firm has weakly lower productivity type and rationally expects its rival to remain whenever it continues with positive probability, so this firm’s payoffs only depend on future states in which both firms survive. That is, evaluating Th,l only requires the continuation value being calculated and on wE(ιi + ιj, ·, j) for all (i, j) 6= (h, l) such that i ≥ h, j ≥ l. Since Procedure1 proceeds in descending order of (h, l), these post-entry values are in hand for the computation of wE(ιl + ιh, ·, l).

16 Return w , w , No Yes E S h = 1? αE, and αS(ιh + STOP

ιl, ·, l) for l < h.

{wE(ιh + ιk, ·, h); 0 ≤ k < h} ← ﬁxed point of Th No Yes l = 1? {wS(ιh + ιk, ·, h); 0 ≤ k < h} ← 2nd term of Th(wE) START h ← l ← l − 1 α (ι + ι , ·) ← I {w (ι + ι , ·, 1) > ϕ} h − 1 E h 1 E h 1 No wE(ιh + ιl, ·, l) ←

ﬁxed point of Th,l Yes αS(ιh + ιl, ·, l) ← h ← kˇ l ← h l < h? wS(ιh + ιl, ·, l) ← I {wE(ιh + ιl, ·, l) > 0}

2nd term of Th,l(wE) 17

Required Functional Operators (   ) X 0 X 0 0 Th,l(f)(c) = max 0, βE  ΠhiΠljπ(ιi + ιj,C , j) + ΠhiΠljwE(ιi + ιj,C , j) + ΠhhΠllf(C ) C = c i,j (i,j)6=(h,l) ( " ! X 0 X 0 X X 0 Th(f)(c, k) = max 0, βE αS(ιh + ιk, c, k) ΠhiΠkjπ(ιi + ιj,C , i) + ΠhhΠkjf(C , j) + ΠhiΠkjwE(ιi + ιj,C , i) i,j j i>h j

X 0 0 0 + {1 − αS(ιh + ιk, c, k)} Πhiπ(ιi,C , i) + Πhh 1 − αE(ιh + ι1,C ) f(C , 0) i X 0 0 0 0 + Πhi 1 − αE(ιi + ι1,C ) wE(ιi,C , i) + ΠhhαE(ιh + ι1,C )f(C , 1) i>h ! #) X 0 0 + ΠhiαE(ιi + ι1,C )wE(ιi + ι1,C , i) C = c , αS(ιh + ι0, ·, 0) ≡ 0, Π00 ≡ 1 i>h

Procedure 1: Initial Equilibrium Calculations for the Heterogeneous Duopoly Model When l reaches 1, the next step is to compute simultaneously the monopoly payoff for a type h firm and the duopoly payoff for a type h firm facing a type k (for all k < h) rival as the

fixed point of Th. Evaluating its right-hand side requires the value function being computed, the entry rule of a potential entrant facing an incumbent with productivity type no less than h ,the corresponding post-entry values for the incumbent, the survival strategies for rivals with types k < h, and the corresponding post-survival values. Again, previous computations using higher values of h, l determined these before this computation begins. Procedure2 complements Procedure1 by determining the entry strategy for a potential monopolist; and the survival strategy for a firm with weakly better productivity type. All but one of these strategies are pure and refelct the values of entry and continuation as expcted. The survival strategy is mixed when both firms have the same type, the payoff to joint continuation is negative, and the payoff isolated continuation is positive. By construction, the resulting probability of survival lies in (0, 1].

Algorithm 1 (Duopoly Equilibrium Calculation). Compute a candidate equilibrium strategy

(αS, αE) and payoﬀs wS and wE in two steps:

1. Use Procedure1 to compute wE, wS, αS(ιh + ιl, c, l) and αE(ιh + ι1, c) for all h, l ∈ K, l < h and c ∈ [ˆc, cˇ].

2. Use Procedure2 to compute αE(ι1, c) for all c ∈ [ˆc, cˇ] and to compute αS(ιh + ιl, c, h) for all h ∈ K, l ∈ {0, . . . , h}, and c ∈ [ˆc, cˇ];

We can use Lemma1 to prove that the constructed equilibrium is unique among all equilibria that default to inactivity.

Proposition 1 (Equilibrium in Heterogeneous Duopoly Model). There exists a unique natural Markov-perfect equilibrium. Algorithm1 computes its payoﬀs and strategy. The equilibrium payoﬀs vS = wS and vE = wE. The equilibrium strategy (aS, aE) = (αS, αE).

Proof. See AppendixA.

3.4 Application

We apply our heterogenous-duopoly model to the welfare analysis of an R&D race game.

Consider a market for some new good. In period t, Ct consumers populate the market. All of these consumers have the same utility function, which is quadratic in the quantity of the new good consumed. Consequently, total demand for the new good at time t and price p equals

Ct(a − p)/b, for some parameters a, b > 0. A ﬁrm supplying q units of this good receives a surplus Ctpq.

18 START

Specify c ∈ [ˆc, cˇ], h ∈ K, and l ∈ {0, 1, . . . , h}

Get wE and wS from Procedure1

Yes αS(ιh + ιl, c, h) ← h > l ? I {wS (ιh + ιl, c, h) > 0}

Yes α (2ι , c, h) ← 1 wS(2ιh, c, h) No S h > 0? h = 1 ?

No Yes

αE(ι1, c) ← Yes wS(ιh, c, h) αS(2ιh, c, h) ← 0 I {w (ι , c, 1) > ϕ} = 0? E 1

wS(ιh, c, h) αS(2ιh, c, h) ← . (9) wS(ιh, c, h) − wS(2ιh, c, h)

STOP

Procedure 2: Calculation of Candidate Survival Rule for the Heterogeneous Duopoly Model

19 Firms must invent the good before they can supply it to the market. This requires that they enter the market, incurring an entry cost ϕ, and subsequently invest in R&D, at a fixed cost κ(k). There are several milestone stages in the invention process, marked by 1, 2,..., kˇ. New entrants start in stage 1 and, as long as they stay in the market and pay the fixed cost κ(k) according to their current stage k, progress through the successive R& D stages according to a Markov chain with transition matrix Π. Once a firm reaches the final stage kˇ, it has invented the product and can start selling it in the market. The fixed cost κ(kˇ) still needs to be paid to produce the good.8 An active firm may exit the market in any stage of the R&D race to avoid paying future fixed costs. We assume that at most two firms are active in the market at any give time. If only one firm is active in stage kˇ, it sells the good at the monopoly price. If two firms are selling the good, they set symmetric quantities to maximize qop + λqrp, where qo and qr denote the quantities set by the firm and its rival, respectively, and λ indexes the level of collusion. If λ = 0, these two firms are Cournot competitors. At higher values of λ, they collude more. If λ = 1, then they operate as if they are branches of a monopoly firm that split their joint monopoly revenues evenly. This game embodies Fershtman and Pakes’(2000) key “semi-collusion” assumption that firms may collude in setting quantities (or prices) but not when choosing R&D investment. Unlike Fershtman and Pakes, we take the level of collusion as exogenously given and ignore the intensive margin of the firms’ strategic R&D investments. This focus on the (entry and exit) decisions to participate in the R&D race allow us to apply the heterogenous-duopoly model to analyze industry dynamics and welfare under different levels of collusion. We find that the model is sufficiently rich to replicate one of Fershtman and Pakes’ main findings: Consumers may benefit from collusion, unlike in static models that take the industry structure as given. Intuitively, the direct negative effect of collusion on consumer welfare through weakened competition in the product market, well known from static models, is counteracted by a positive effect on R&D participation that increases product availability and product market competition. To obtain this result, we first compute the model’s unique natural Markov-perfect equilibrium for each value of λ between 0 and 1, with a 0.01 increment. Throughout, we specify Q(·|C) to approximate a random walk in the logarithm of C with innovation variance 0.32, reflected off of the state space’s upper and lower boundaries, lnc ˆ = −1.5 and lnc ˇ = 1.5. Also, we specify kˇ = 4, β = 0.95, κ(k) = 20 for all k, ϕ = 470, demand parameters a = 20 and b = 2, and the Markov transition matrix Π for the R&D stages so that firms either progress ˇ one stage or remain put: Πk,k = Πk,k+1 = 0.5 for all k < k and Πk,ˇ kˇ = 1.

8Alternatively, we can let the ﬁxed cost decline with k. This only requires a minor adjustment of the model.

20 Subsequently, for each value of λ, we use the equilibrium strategy to simulate the market’s evolution over 100 periods, starting from a fixed c0 = 2.718, drawn from the demand process’s ergodic distribution, and an empty market. We repeat the simulation 10,000 times, drawing new demand and type transitions in each simulation, but using the same random draws across the different values of λ. To analyze the impact of collusion on welfare, we compute, for each level of collusion λ, the discounted sum FP (λ) of all firms’ revenues net of all firms’ fixed costs and entry costs over the 100 periods, and the discounted sum CS(λ) of the consumer surplus over the 100 periods, both averaged over the 10,000 simulation runs. We assume that consumers have the same discount factor as firms. The total surplus TS(λ) ≡ FP (λ)+CS(λ).

Figure 5: Welfare Analysis for Various Levels of Collusion

Consumer Surplus Firm Profit Total Surplus 0.7 0.8 1.2

0.6 0.7 1.1 )/TS(0) )/TS(0) 0.5 )/TS(0) 0.6 λ λ λ 1 TS( FP( CS( 0.4 0.5

0.4 0.9 0 0.5 1 0 0.5 1 0 0.5 1 λ λ λ Average Number of Average Number of Active Firms Periods before Success 2 6.5

1.5 5.5 #Firms #Periods 5

1 4.5 0 0.5 1 0 0.5 1 λ λ The upper-left and upper-middle panels of Figure5 show CS(λ) and FP (λ) for each value of λ, as a proportion of the competitive market’s total surplus TS(0). First, if λ increases from 0, CS(λ) gradually increases and FP (λ) gradually decreases. Then, CS(λ) jumps up and FP (λ) jumps down. At higher levels of collusion, increases in λ decrease the consumer surplus and increase firms’ profits. Clearly, for low values of λ, the positive effect of increasing collusion on R&D investment dominates its direct weakening effect on product market competition. Figure5’s bottom-left panel sheds further light on this. It plots the number of active firms for each λ, averaged over the 100 periods and all the simulation runs. We observe a gradual increase and then

21 an upward jump in the number of firms, paralleling the increase and jump in the consumer surplus. If λ is low; that is, with little or no collusion; no entrant facing a monopoly market can recover the sunk cost of entry, even when demand is at its highest level. Therefore, markets with little collusion are often monopoly markets. If λ increases, firms expect higher payoffs from a duopoly product market, and are more willing to participate in the R&D race, even if one firm is already in this race. The value of λ at which the number of firms and welfare jump is the level of collusion above which two firms enter immediately, in the initial demand state c0. This increase in the number of firms improves the consumer surplus in two ways. First, it improves product availability. Specifically, in this example, on average the first product reaches the market faster with higher levels of collusion (see Figure5’s bottom-middle panel). Second, it mitigates the anticompetitive effects of collusion, by ensuring that consumers are more often charged the (collusive) duopoly price, which, for all λ < 1, is lower than the monopoly price. At low levels of λ, the consumer welfare enhancing effects dominate the direct negative effects of increased collusion. In contrast, as is clear from Figure5’s bottom panels, at higher levels of collusion, the market is often served by the maximum number of two firms. Consequently, further increases in λ have only small effects on the number of firms serving the market and the speed at which the good becomes available. Therefore, at higher levels of collusion, the direct effects of collusion dominate, and the consumer surplus gradually falls if λ increases. Nevertheless, the benefits from earlier consumption under full collusion (λ = 1) ensure that CS(1) > CS(0). The variation of FP (λ) with λ mirrors the variation of CS(λ). If λ crosses the level at which two firms immediately enter the market, instead of one, the total fixed cost incurred is doubled, but the total revenue is not. Consequently, FP (λ) jumps down. For similar reasons, FP (λ) falls gradually if λ increases at lower levels of collusion. In contrast, at higher levels of collusion, the market is usually a duopoly and the market structure does not change much with increases in λ. Consequently, the positive effects of such increases on the collusive duopoly price dominate, and FP (λ) increases. Finally, FP (1) < F P (0), because of scale savings: The monopoly price is usually charged at either collusive extreme, but two firms, instead of one, often incur fixed costs under full collusion. Figure5’s upper-right panel plots the total surplus as a fraction of the competitive market’s total surplus. At low levels of collusion, an increase in λ increases TS(λ). In particular, the upward jump in the number of firms leads to an upward jump in the total surplus. At these levels of collusion, the positive effects of increased product market competition and earlier consumption on consumer welfare dominate its negative effects on firms through price decreases and fixed cost increases. At higher levels of collusion, the total surplus falls with increases in λ, because R&D activity is hardly affected and the negative welfare effects of

22 collusion familiar from static models dominate. In this specific example, as in static models that take the market structure as given, full collusion in the product markets lowers welfare below that in a competitive market: TS(1) < T S(0). However, unlike in such static models, the competitive market is often served by only one firm and monopolistic pricing is common under both levels of collusion. Consequently, the result that full collusion lowers total welfare cannot be explained by the usual negative welfare effects of collusive pricing. Instead, it is due to the waste of fixed costs caused by excess entry of producers, which is not offset by the gains from earlier consumption.9 It is worth stressing that these results are obtained at a very low computational cost. For any particular value of λ, with 301 grid points for C and the parameter values of this section’s experiment, we can solve the model within one second using Matlab on a PC. Even with β = 0.995 (monthly data) and kˇ = 10, which implies a state space with over 33,000 points, we can solve the model in about 5–30 seconds on a PC.10 This feature of our framework makes it a very useful complement to existing richer, but computationally more forbidding, frameworks for the analysis of industry dynamics. For example, Fershtman and Pakes’ framework allows for a more detailed study of collusion dynamics by modeling, among other things, the intensive margin of R&D investment and endogenizing collusion. However, their framework’s comparative richness comes at a substantial computational cost: It makes the replication of their results across different parameter values very hard. In contrast, our framework allows us to quickly examine the welfare implications of collusion for a wide range of parameter values.

4 The General Model

We now turn to the general model, with arbitrary finitem ˇ and kˇ. The central difficulty of the equilibrium analysis is that the equilibrium payoff function does not necessarily satisfy a monotonicity property analogous to the ones in Lemmas1. In Section 4.1, we first analyze a type of equilibrium in which the payoffs still retain the monotonicity property: Adding an active firm into the industry weakly decreases other firms’ payoffs. We can straightforwardly extend Algorithm1 and use a sequence of contraction mappings to efficiently compute such an equilibrium, if it does exist. This monotonicity property is the key to establish the essential

9Obviously, this result can be reversed if consumers are impatient and/or have much larger weight in the total surplus than producers do. 10We use value function iteration to compute the ﬁxed points of the contraction mappings, which simpliﬁes our code, but results in a (slow) linear convergence rate in β. To cope with this issue, one can turn to more sophisticated approaches (see Judd, 1998, for a brief survey). For example, Ferris, Judd, and Schmedders’s (2007) Newton-based method ensures global convergence with a quadratic convergence rate.

23 uniqueness of this type of equilibrium. However, since the monotonicity property does not always hold in the general model, this type of equilibrium may not exist. In Section 4.2, we discuss a simple example in which the monotonicity of equilibrium payoffs is violated and multiple equilibria emerge. In one class of those equilibria, if firms were allowed to renegotiate, they could strictly improve their payoffs by playing another equilibrium. We continue to focus on the type of equilibria that are renegotiation-proof and establish their existence. An extension of our algorithm can compute all such equilibria, if C has a discrete distribution.

4.1 Payoﬀ-Monotone Equilibrium

We define an equilibrium to be payoff-monotone if the equilibrium payoffs satisfy conditions analogous to the ones in Lemma1.

Definition 3. A Markov-perfect equilibrium is payoff-monotone if its equilibrium payoff functions satisfies vS(m, c, k) ≥ vS(m+ιk, c, k) and vE(m, c, k) ≥ vE(m+ιk, c, k) for all (m, c, k).

We showed in Section3 that duopoly firms of the same type choose to continue if continuation guarantees a positive payoff, because the heterogenous duopoly model’s equilibrium payoffs satisfy Lemma1. This property allows us to construct a sequence of contraction mappings in Algorithm1 to compute the unique natural Markov-perfect equilibrium. Similarly, in the general model, suppose that, for some parameter values, there exists a payoff-monotone natural Markov-perfect equilibrium. Then, if continuation renders the payoff to all firms of the same type positive, continuation is the dominant strategy for these firms. Following the argument leading to condition (5) in Section 3.1, we can establish similar necessary conditions on equilibrium payoffs. For instance, in the market withm ˇ type kˇ firms, the payoff-monotone ˇ equilibrium payoff vE(mι ˇ kˇ, c, k) necessarily satisfies

ˇ 0 ˇ 0 ˇ vE(mι ˇ kˇ, c, k) = max{0, βE[π(mι ˇ kˇ, c , k) + vE(mι ˇ kˇ, c , k)|C = c]}. (10) ˇ The right hand side of (10) defines a Bellman operator that uniquely determines vE(mι ˇ kˇ, ·, k). Note that the heterogeneous duopoly model and the general model only differ in the number of firms and share essentially the same dynamic specification. Therefore, Algorithm1 can be naturally extended to solve for the payoff-monotone natural equilibrium, by computed the fixed points of a sequence of contraction mappings. Similarly to Algorithm1, we partition the state space, order the parts, and compute the equilibrium in a corresponding sequence of steps. Each step covers the computation on a single part of the state space. We order the steps so that all results that are needed in later steps are passed on from earlier steps. The partition and its order are defined using an oriental lexicographic order.

24 Deﬁnition 4. Oriental lexicographical superiority (OLS) is a relation over Rn. For any n pair of vectors x, y ∈ R , x y if xn > yn, or (xn = yn and xn−1 > yn−1), or ,..., or (xn = yn and xn−1 = yn−1 and ... and x1 > y1).

We use the phrase “oriental” because the vectors x, y are read from right to left when being compared, as in Arabic and Hebrew. In the previous sections, we have implicitly used an ordering based on OLS; the equilibrium payoff for an OLS market structure is always computed before the payoffs in any state it is superior to. For example, in Section 3.1’s one productivity type example, is equivalent to > on R and the payoff to a duopolist is computed first, followed by the payoff to a monopolist. In Section 3.2, the sequence of market structures considered was {2ιH, ιH + ιL, ιH, 2ιL, ιL}. Thus, we partitioned the state space into five parts and ordered them in decreasing OLS to compute the equilibrium payoffs and strategy. Furthermore, this ordering extends to Algorithm1 as well; the index pair ( h, l) in Procedure1 is decreasing in OLS. This ordering ensures that equilibrium payoffs and entry/survival rules necessary for computation in later steps are calculated in earlier steps. We construct the algorithm for the general model following the same ordering. For any ˇ ˇ (m, ˇ kˇ) pair, there are mˇ +k − 1 = (m ˇ +k)! − 1 possible non-empty markets. First, we partition kˇ kˇ!m ˇ ! ˇ the state space into (m ˇ +k)! −1 parts, with each step of the algorithm computing the payoff on kˇ!m ˇ ! one of these parts. In step i, to see what the states in this part are, suppose the i-th ranked i i i market structure in the OLS sequence is m . Let k = min{k ∈ K; mk > 0} be the lowest i type of active firm in m and let a set Mmi collect all the market structures that share the same number of type-ki, ki + 1,..., kˇ firms as mi. The part of the state space considered in i this step is then {(m, c, k ); m ∈ Mmi , c ∈ [ˆc, cˇ]}. In other words, in the i-th step, we compute i a type-k firm’s payoff in every market structure in Mmi for all c. Since this part of the state space are constructed from mi, we say that it is indexed by mi, and hence name mi as the indexing market structure.

The ﬁrst step of the algorithm is indexed by the most superior market structure,mι ˇ kˇ. Then, we proceed the algorithm and sequentially determine the payoﬀs and strategy for market structures in the order of decreasing OLS.

25 Algorithm 2 (Calculation of a Candidate Equilibrium for the General Model).

START

ˇ βπ(ιkˇ, c,ˇ k) mˇ ← max{n ∈ ; π(ιˇ + (n − 1)ι , c,ˇ kˇ) + > 0} N k 1 1 − β

αS(·) ← 1, αE(·) ← 0, wE(·) ← 0

ˇ Order all elements from the set {m ∈ Zk; 1 ≤ |m| ≤ mˇ } by . Index the obtained sequence by m1, m2,....

ˇ i ← (m ˇ +k)! − 1 kˇ!m ˇ !

i i k ← min{k ∈ K; mk > 0}  ki−1 ki−1   i X i X  Mmi ← m + ιkmk; |m | + mk ≤ mˇ  k=1 k=1  i i HS ← (m, c, k); m ∈ Mmi , k = k

i i i For all HS ∈ HS, compute wE(HS) as the ﬁxed point of i 0 0 0 i0 i (T f)(HS) = max {0, βE [π(N ,C ,K ) + g(HS ) HS]} , i 0 0 0 i0 i and wS(HS) = βE [π(N ,C ,K ) + g(HS ) HS] , with  i0 i0 i  f(HS ) if HS ∈ HS i0 ˇ g(HS ) = (m ˇ +k)! −1 i0 i0 i+1 S S kˇ!m ˇ !  wE(HS ) if HS ∈ HS ... HS

mi = 1? No ki i ← i − 1 or ki = 1?

Yes Compute survival/entry rule.

No i = 1?

Yes

STOP

26 In Algorithm2, when computing the candidate post-entry payoff wE as the fixed point of the Bellman equation, the expectation relies on the relevant parts of other firms strategy11. Because of the algorithm’s OLS ordering, for the firms with lower productivity types than the firm of interest and the potential entrants, these values have been computed in previous steps. The survival rules for firms with productivity types at least as good as the firm of interest are set to continuation. Also, the OLS ordering of the algorithm helps to ensure that the current states that the firm of interest is facing only evolves to states that have been covered in previous computation, providing that this firm continues. Hence, all relevant values of future post-entry payoff have been computed in previous steps. Then, T is always a contraction mapping with unique fixed point wE(HS). Procedure3 is devoted to computing the survival/entry strategy. In this procedure, when firms are randomizing between survival and exit, the mixing probability is chosen to be one of possible probabilities that solves the indifference condition (11). If it is not profitable to unilaterally deviate from exit to survival given all other firms of the same type opt for exit, the mixing probability can also be set to 0.

Note that Algorithm2 does not require wE to be monotone as in Definition3. After computing wE with Algorithm2, we can check whether it satisfies this monotonicity condition. If it does, we can show that the candidate equilibrium strategy (αS, αE) is unique. We can also verify that (αS, αE) forms a natural Markov-perfect equilibrium. Since the Bellman equation for wE defines the necessary condition for any payoff-monotone natural Markov-perfect equilibrium payoff, if one such equilibrium exists, not only we are able to compute it using Algorithm2, but also we can prove its essential uniqueness.

Proposition 2 (Payoﬀ-Monotone Equilibrium in the General Model). If there exists a payoﬀ- monotone natural Markov-perfect equilibrium, it is the unique such equilibrium and Algorithm

2 computes it. The post-entry equilibrium payoﬀ function is wE and the equilibrium strategy is (αS, αE).

Proof. See AppendixB.

4.2 Renegotiation-Proof Equilibrium

Proposition2 implies that a payoff-monotone natural Markov-perfect equilibrium does not exist if wE is not monotone as in Definition3. In AppendixC, we present a simple example in which equilibrium payoffs are not monotone and there are multiple natural Markov-perfect equilibria. In this example, we consider an industry with at most three active firms. We assume that firms can be type H or type

11Computing the market structure transition matrix conditional on ﬁrms’ strategy is conceptually straightforward, but practically involved. We describe the details in Section ?? of the online appendix.

27 mi, ki

i Specify c ∈ [ˆc, cˇ] and m ∈ i , ∀0 ≤ j ≤ mˇ − |m | Mm +jιki

i Yes αS(m, c, k ) ← i mk = 1 ? i I[wE(m, c, k ) > 0]

i Yes wE(m, c, k ) α (m, c, ki) ← 1 > 0? S

No Find all p’s ∈ [0, 1) satisfying, m −1 ki m i − 1 X m i −1−j j k i (1−p) k p w (m−(m i −1−j)ι i , c, k ) = 0. (11) j S k k j=0

i i No wS(m , c, k ) Yes > 0?

i i αS(m, c, k ) ← 0 αS(m, c, k ) ← or one of the p’s one of the p’s

α (m, c) ← I{w(m + E Yes i jι1, c, 1) − ϕ > k = 1 ? 0, ∀0 ≤ j ≤ mˇ − |m|}

CONTINUE

Procedure 3: Calculation of Candidate Entry/Survival Rule for the General Model

28 L. For two type H duopoly firms contemplating survival, we create a situation that if these two firms jointly continue to next period, any type L potential entrant will never find it profitable to enter this market. This way, the type H duopolists deter any future entry by joint survival and enjoy a high duopoly surplus forever. Otherwise, if one of the firms exits, then two type-L firms will enter the market and remain active onwards. The survived type-H firm will only receive a low triopoly surplus thereafter. Connecting this example to the static survival game depicted in Figure2, we construct the payoff matrix such that, for some c, the post-survival value satisfies vS(2ιH, c, H) > 0 > vS(ιH, c, H). Therefore, although “Sur- vive, Survive” remains an equilibrium in this static game, “Exit, Exit” emerges as another equilibrium. Also, there could be equilibria involving mixed strategies. Indeed, we show in AppendixC that we do have three possible equilibrium actions at this particular point of the game tree. Namely, to survive for sure, to exit for sure, and to survive with some probability. We further demonstrate that when three firms are randomizing between survival and exit because joint survival is not profitable, the mixing probability can be multiple. We distinguish two sources of equilibrium multiplicity using this example. One comes from the incumbents’ failure to jointly continue if this is profitable. If the two type-H firms can coordinate on continuation, they can strictly improve their equilibrium payoffs. Since these two firms repeatedly interact, it seems reasonable to assume that they are able to “renegotiate” to joint continuation whenever this is profitable. Henceforth, we restrict attention to equilibria with the desirable property that firms cannot improve their payoffs by one-shot change of action. We call this property renegotiation-proofness.

Deﬁnition 5. A natural Markov-perfect equilibrium is (one-shot) renegotiation-proof if, for any (m, c) pair, no one-shot agreement satisfying the following properties can be negotiated:

• all ﬁrms in the agreement change their survival actions once;

• the agreement is self-enforcing, so no ﬁrm in the agreement has incentive to unilaterally change the agreed action;

• if one type k ﬁrm is in the agreement, all type k ﬁrms are; and

• the payoﬀs to all ﬁrms in the agreement are strictly improved.

In any equilibrium, a firm earns positive payoffs only when continuing for sure. Therefore, if all firms of a certain type can strictly improve their payoff by changing their actions, it must be the case that (i) the actions must be changed from exiting with non-negative probability to surviving with probability one (ii) the actions of joint continuation must give all firms in the agreement positive payoff. Therefore, this refinement has bite only when all incumbents of certain type(s) could coordinate on sure joint continuation and earn positive payoffs, but will

29 not unilaterally continue if others do not. Note that in the duopoly model, Lemma1 ensures that both incumbents of the same type continue for sure if joint continuation renders payoff positive. Therefore, no further improvement is possible via renegotiation. Consequently, the natural equilibrium in Section3 is renegotiation-proof. Since the monotonicity in Definition 3 essentially functions in the same way as the monotonicity in Lemma1, the payoff-monotone equilibrium in the general model is also renegotiation-proof.

Recall that in Algorithm2, αS(m, c, k) is set to its initial value of one when computing wE(m, c, k). This implies that all type-k ﬁrms are “forced” to jointly continue if positive payoﬀ is expected. Therefore, the Bellman equation for wE is a necessary condition on wE for a renegotiation-proof natural Markov-perfect equilibrium. When verifying that (αS, αE) forms a natural equilibrium, we also verify that it is a renegotiation-proof one. We can further show that Algorithm2 always delivers some ( αS, αE) as its outcome, which proves the existence of a renegotiation-proof natural Markov-perfect equilibrium.

Proposition 3 (Renegotiation-proof Equilibrium in the General Model). Algorithm2 always computes some (αS, αE) and this strategy (αS, αE) forms a renegotiation-proof natural Markov-perfect equilibrium. So, a renegotiation-proof natural Markov-perfect equilibrium always exists.

Proof. See AppendixB.

The renegotiation-proof property helps to eliminate the equilibria involving exit and mixing actions when joint continuation is profitable. However, the other source of the multiplicity persists. As we have illustrated in the example in AppendixC, when joint survival is not profitable and more than two firms are randomizing between survival and exit, there can be multiple equilibrium mixing probabilities. The property of renegotiation-proofness is silent on which probability to select. Therefore, each distinct equilibrium mixing probability leads to a different equilibrium survival rule. Different combinations of the equilibrium survival rules result in different renegotiation-proof natural Markov-perfect equilibria. Since the Bellman equation for wE defines the necessary condition for renegotiation-proof natural Markov-perfect equilibrium payoff, any such equilibrium should be an outcome of Algorithm 2, if the mixing probabilities that correspond to this equilibrium are used in the computation. Recall that we do have proven in Proposition2 that if a payoff-monotone equilibrium exists, the mixing probability is always unique, and such equilibrium is the unique outcome of Algorithm2. This implies the following corollary to Proposition2.

Corollary 1. If there exists a payoﬀ-monotone natural Markov-perfect equilibrium, it is also the unique renegotiation-proof natural Markov-perfect equilibrium.

30 If there is no payoff-monotone equilibrium, the possible multiplicity of mixing probabilities may challenge our equilibrium computation. After all, each step of Algorithm2 requires the unique input of payoffs and rules computed in the previous steps. (In Section 4.1, Algorithm 2 simply selects an arbitrary mixing probability to continue when the multiplicity arises.) In Section ?? of the online appendix, we prove that the number of renegotiation-proof natural Markov-perfect equilibria is finite if C is discrete. We also extend Algorithm2 so that it computes all renegotiation-proof natural Markov-perfect equilibria, by creating parallel branches of Algorithm2 every time the multiplicity arises, with each branch corresponding to a distinct choice of mixing probability.

References

Abbring, J. H., and J. R. Campbell (2010): “Last-In First-Out Oligopoly Dynamics,” Econometrica, 78(5), 1491–1527.1

Abbring, J. H., J. R. Campbell, and N. Yang (2010): “Sunk Costs, Entry and Exit in Dynamic Oligopoly,” Mimeo, CentER, Department of Econometrics & OR, Tilburg University.1

Bresnahan, T. F., and P. C. Reiss (1990): “Entry in Monopoly Markets,” Review of Economic Studies, 57(4), 531–553.1

Cabral, L. M. (1993): “Experience Advantages and Entry Dynamics,” Journal of Eco- nomic Theory, 59, 403–416.2,6

Doraszelski, U., and A. Pakes (2007): “A Framework for Applied Dynamic Analysis in IO,” in Handbook of Industrial Organization, ed. by M. Armstrong, and R. H. Porter, vol. 3, chap. 4. Elsevier Science, Amsterdam.1

Ericson, R., and A. Pakes (1995): “Markov-Perfect Industry Dynamics: A Framework for Empirical Work,” Review of Economic Studies, 62, 53–82.1,2,5

Ferris, M., K. Judd, and K. Schmedders (2007): “Solving Dynamic Games with New- ton’s Method,” Discussion paper, University of Wisconsin, Madison. 23

Fershtman, C., and A. Pakes (2000): “A Dynamic Oligopoly with Collusion and Price Wars,” RAND Journal of Economics, 31(2), 207–236.0,2, 20, 23

Fudenberg, D., and J. Tirole (1991): Game Theory. MIT Press, Cambridge, MA.6

Judd, K. L. (1998): Numerical Methods in Economics. MIT Press. 23

31 Appendices

A Proofs for Section3

Proof of Lemma1. First, we verify a property of the post-entry payoff function. ˇ Property 1. For a given k ∈ K, and for all x such that 0 ≤ x ≤ k, if vE(ιk + ιx, ·, k) is weakly decreasing in x, we say that vE satisfies Property1 for k. Intuitively, Property1 requires that a type- k firm’s post-entry payoff is weakly decreasing in its opponent’s type. A sufficient condition for Lemma1 is that in any natural equilibrium, vE satisfies Property1 for all k ∈ K. In order to prove this sufficient condition, we also need to verify vE satisfies three other interdependent properties for all k ∈ K. ˇ Property 2. For a given k ∈ K, and for all x such that 0 ≤ x ≤ k, if vE(ιk + ιx, ·, x) is weakly increasing in x, we say that vE satisfies Property2 for k.

Property 3. For a given k ∈ K, and for all x such that 1 ≤ x ≤ k − 1, if vE(ιk + ιx, ·, x) ≤ vE(ιk−1 + ιx, ·, x), we say that vE satisﬁes Property3 for k.

Property 4. For a given k ∈ K, if vE(ιk, ·, k) ≥ vE(ιk−1, ·, k − 1), we say that vE satisﬁes Property4 for k12. To prove the three properties for all k, we ﬁrst show that given any equilibrium strategy

(aS, aE), vE is computed as the unique fixed point of a contraction mapping. Then, we prove ˇ in turn that vE satisfies Property2,1,3, and4 for k = k. Therefore, if we have a Banach space in which all elements are functions that satisfy the properties for kˇ, the contraction maps this space into itself. Next, we iterate on k = kˇ −1,..., 1. In each step, we start with a Banach space in which all elements are functions that satisfy the properties for k + 1,..., kˇ. We know from the previous steps that the contraction maps the space into itself. Then, we construct a smaller Banach space by requiring all its elements to additionally satisfy the properties for k. We further show that the contraction maps the smaller Banach space into itself. This procedure gives us a sequence of shrinking spaces, with the smallest one contains all functions that satisfy Properties1-4 for all k. Because vE is the unique fixed point of the contraction, it must be in the smallest space and satisfy these properties for all k. We leave all the algebraic details to Section ?? of the online appendix.

After Property1 is veriﬁed for all k, vE(2ιk, c, k) ≤ vE(ιk, c, k) for any k ∈ K follows immediately. Using the deﬁnition in equation (2) and Property1, we can prove vS(2ιk, c, k) ≤ vS(ιk, c, k) for any k ∈ K. The details are in the online appendix.

12 For completeness, vE(ι0, ·, 0) ≡ 0.

32 Proof of Proposition1. We prove the proposition in two steps. First, we establish a lemma verifying that the candidate equilibrium computed by Algorithm1 is indeed a natural Markov-perfect equilibrium. Then, we use Lemma1 to prove that the constructed equilibrium is essentially unique.

Lemma 2. The strategy (αS, αE) and payoﬀ function wE constructed by Algorithm1 form a natural Markov-perfect equilibrium.

Proof. The proof for Lemma2 has two parts. First, note that Algorithm1 already embodies 1 2 1 the requirement in Definition1, i.e., for k > k , holding αS(ιk1 + ιk2 , c, k ) = 1 when computing the payoff and strategy for k2 firm. We then need to verify that the candidate 1 equilibrium payoff function wE supports this heuristic, i.e., wE(ιk1 + ιk2 , c, k ) > 0 whenever 2 αS(ιk1 +ιk2 , c, k ) > 0. Second, we show that (αS, αE) forms a natural equilibrium by proving that it is one-shot-deviation-proof.

1 2 Prove wE(ιk1 + ιk2 , ·, k ) ≥ wE(ιk1 + ιk2 , ·, k ). Given any (αS, αE), wE is computed as the ﬁxed points of the contraction mappings in Algorithm1. This enables us to use the same trick as in the proof for Lemma1. We focus on a Banach space in which all elements 1 2 are functions that satisfy f(ιk1 + ιk2 , ·, k ) ≥ f(ιk1 + ιk2 , ·, k ). Then we prove that all the contractions computing wE map the space into itself. The algebraic details are in the online appendix, Section ??.

Verify one-shot deviation proofness. To verify one-shot deviation proofness for αS, we need to show that for any k1, k2, c, 1 ≤ k1 ≤ kˇ and 0 ≤ k2 ≤ kˇ,

1 0 1 αS(ιk1 + ιk2 , c, k ) ∈ arg max aE[wS(M , c, k )|M = ιk1 + ιk2 ] (12) a∈[0,1] where

 2 1 a (ι 1 + ι 2 , c, k )w (ι 1 + ι 2 , c, k )  S k k S k k 0 1 2 1 1 2 E[wS(M , c, k )|M = ιk1 +ιk2 ] = +(1 − aS(ιk1 + ιk2 , c, k ))wS(ιk1 , c, k ) if k ≥ k (13)  1 1 2  wS(ιk1 + ιk2 , c, k ) if k < k and wS is deﬁned analogously as vS by equation2. To verify (12), consider the following cases

1 1 1. For all c such that αS(ιk1 + ιk2 , c, k ) = 1, we know wE(ιk1 + ιk2 , c, k ) > 0. Then, we 0 1 show that E[wS(M , c, k )|M = ιk1 + ιk2 ] > 0.

1 2 1 1 (a) If k ≤ k , then wE(ιk1 + ιk2 , c, k ) is computed by Tk2,k1 and wE(ιk1 + ιk2 , c, k ) = 1 0 1 max {0, wS(ιk1 + ιk2 , c, k )} > 0. Then, from (13), E[wS(M , c, k )|M = ιk1 +ιk2 ] = 1 wS(ιk1 + ιk2 , c, k ) > 0.

33 1 2 1 1 (b) If k > k , then wE(ιk1 + ιk2 , c, k ) is computed by Tk1 and wE(ιk1 + ιk2 , c, k ) = 0 1 0 1 max {0, E[wS(M , c, k )|M = ιk1 + ιk2 ]} > 0, so E[wS(M , c, k )|M = ιk1 +ιk2 ] > 0.

0 1 1 So, arg maxa∈[0,1] aE[wS(M , c, k )|M = ιk1 + ιk2 ] = {1} 3 αS(ιk1 + ιk2 , c, k ) = 1.

1 1 2. For all c such that αS(ιk1 + ιk2 , c, k ) = 0, we know wE(ιk1 + ιk2 , c, k ) = 0. Then, we 0 1 show that E[wS(M , c, k )|M = ιk1 + ιk2 ] = 0.

1 2 1 1 (a) If k < k , then wE(ιk1 + ιk2 , c, k ) is computed by Tk2,k1 and wE(ιk1 + ιk2 , c, k ) = 1 0 1 max {0, wS(ιk1 + ιk2 , c, k )} = 0. Then, from (13), E[wS(M , c, k )|M = ιk1 +ιk2 ] = 1 wS(ιk1 + ιk2 , c, k ) ≤ 0.

1 2 2 (b) If k ≥ k , then in natural equilibrium it must be that αS(ιk1 + ιk2 , c, k ) = 0, and 0 1 1 hence (13) gives E[wS(M , c, k )|M = ιk1 + ιk2 ] = wS(ιk1 , c, k ). Because wE(ιk1 + 1 1 1 ιk2 , c, k ) is computed by Tk1 and wE(ιk1 + ιk2 , c, k ) = max {0, wS(ιk1 , c, k )} = 0, 0 1 E[wS(M , c, k )|M = ιk1 + ιk2 ] ≤ 0.

0 1 1 So, arg maxa∈[0,1] aE[wS(M , c, k )|M = ιk1 + ιk2 ] 3 αS(ιk1 + ιk2 , c, k ) = 0. Note that if 0 1 we require default to inactivity, arg maxa∈[0,1] aE[wS(M , c, k )|M = ιk1 + ιk2 ] = {0}.

1 1 2 3. For all c such that αS(2ιk1 , c, k ) is determined by (9), then k = k and

0 1 1 1 1 1 E[wS(M , c, k )|M = ιk1 +ιk2 ] = αS(2ιk1 , c, k )wS(2ιk1 , c, k )+(1−αS(2ιk1 , c, k ))wS(ιk1 , c, k ) = 0.

0 1 The last equality is due to equation (9). So, arg maxa∈[0,1] aE[wS(M , c, k )|M = ιk1 + 1 ιk2 ] = [0, 1] 3 αS(2ιk1 , c, k ).

To verify one-shot-deviation-proofness for αE, we need to show that for any k, c, 0 ≤ k ≤ kˇ,

0 αE(ιk + ι1, c) ∈ arg max a(E[wE(M , c, 1)|M = ιk + ι1] − ϕ) (14) a∈[0,1] where ( 0 wE(ιk + ι1, c, 1) if k > 0 E[wE(M , c, 1)|M = ιk+ι1] = (1 − αE(2ι1, c))wE(ι1, c, 1) + αE(2ι1, c)wE(2ι1, c, 1) if k = 0

By construction, αE(ιk + ι1, c) satisﬁes (14) except for k = 0. Thus, at this moment, we can assert that αE(ιk + ι1, c) is a natural Markov-perfect equilibrium strategy for all k > 0.

Since no operator T in Algorithm1 depends on αE(ι1, c), this result is sufficient to ensure that the fixed points, wE is the natural Markov-perfect equilibrium payoff corresponding to

(αS, αE). Lemma1 then guarantees that wE also exhibits wE(2ι1, c, 1) ≤ wE(ι1, c, 1). Thus,

34 1. when αE(2ι1, c) = 1, it must be the case that wE(ι1, c, 1) ≥ wE(2ι1, c, 1) > ϕ so

αE(ι1, c) = 1. The right-hand-side of (14) is

0 arg max a(E[wE(M , c, 1)|M = ιk+ι1]−ϕ) = arg max a(wE(2ι1, c, 1)−ϕ) = {1} 3 αE(ι1, c). a∈[0,1] a∈[0,1]

0 2. when αE(2ι1, c) = 0, E[wE(M , c, 1)|M = ιk + ι1] = wE(ι1, c, 1). So αE(ι1, c) = I[wE(ι1, c, 1) > ϕ] satisﬁes (14).

So, we conclude that (αS, αE) forms a natural Markov-perfect equilibrium and wE, wS are the associated payoﬀs.

With Lemma2 in hand, we can prove the uniqueness of natural Markov-perfect equi- 1 ˇ librium following the order of Procedure1, starting from k1 = k. First, observe that in a ˇ ˇ ˇ symmetric equilibrium, if vS(2ιkˇ, c, k) ≤ 0, then vE(2ιkˇ, c, k) = 0. Then, if vS(2ιkˇ, c, k) > 0, ˇ ˇ Lemma1 ensures that in any equilibrium, vS(ιkˇ, c, k) ≥ vE(2ιkˇ, c, k) > 0 for any n < mˇ . This ˇ means that continuation dominates any other action when vS(2ιkˇ, c, k) > 0. Therefore, any ˇ equilibrium post-entry payoff must be a fixed point of Tk,ˇ kˇ in Algorithm1, vE(2ιkˇ, ·, k) = ˇ wE(2ιkˇ + ιk, ·, k). Consequently, for any k < k, wE(ιkˇ + ιk, ·, k) is determined as the unique natural Markov-perfect equilibrium payoff. Recall that in the proof for Lemma2, the optimal strategy sets for type-k firm are singletons. Therefore, αS(ιkˇ +ιk, ·, k) and αE(ιkˇ +ι1, ·) is the ˇ unique natural Markov-perfect equilibrium strategy, which guarantees that wE(ιkˇ + ιk, ·, k) is ˇ the unique post-entry equilibrium payoff. The uniqueness of wS(ιkˇ + ιk, ·, k) readily follows ˇ and the uniqueness of aS(2ιkˇ, ·, k) is ensured by equation (9).

In the i-th steps in Procedure1, suppose the pair of types considered is ( h, l) = (k1, k2). When k1 = k2, Lemma1 and equilibrium symmetry again ensure that any equilibrium 1 1 post-entry payoff vE(2ιk1 , ·, k ) must be a fixed point of Tk1,k1 . Hence, vE(2ιk1 , ·, k ) = 1 2 1 wE(2ιk1 , ·, k ) is the unique natural Markov-perfect equilibrium payoff. Then, for any k < k , 2 since Tk1,k2 does not depend on any strategy, wE(ιk1 + ιk2 , ·, k ) is determined as the unique 2 post-entry equilibrium payoff and αS(ιk1 + ιk2 , ·, k ), αE(ιk1 + ι1, ·) as the unique natural Markov-perfect equilibrium strategy that is consistent with payoff-maximization. Because

Tk1 only depends on natural Markov-perfect equilibrium strategy that has been verified to be 1 unique, wE(ιk1 + ιk2 , ·, k ) is then uniquely determined as the post-entry equilibrium payoff. 1 2 The uniqueness of wS(ιk1 +ιk2 , ·, k ) and wS(ιk1 +ιk2 , ·, k ) are then straightforwardly verified. 1 Equation (9) ensures the uniqueness of aS(2ιk1 , ·, k ).

B Proofs for Section4

To prove Propositions3 and2, we prove a useful lemma ﬁrst.

35 Lemma 3. Algorithm2 always delivers some (αS, αE) as outcome. Furthermore, (αE, αS) forms a natural Markov-perfect equilibrium, with payoﬀs wE and wS.

Proof. We follow four steps to prove this lemma. We first show that Algorithm2 computes wE, wS, αS, and αE for all (m, c, k). Second, we prove that Algorithm2 always delivers some well-defined (αS, αE) as outcome. This is a nontrivial step. Because we need to show that wE as the fixed point of T always exist, and αS is well defined when Procedure3 assigns p from equation (11) to it.

After proving the first part of the lemma, we verify in the third step of the proof that αS satisfies the requirement in Definition1. Eventually, we prove that in each step of Algorithm

2, wE is constructed as an equilibrium post-entry value, and the corresponding wS gives the equilibrium post-survival payoﬀ. Along the way, we also show that (αS, αE) is an natural equilibrium strategy. ˇ First, note that the set of all indexing market structures is M ≡ {m ∈ Zk; 1 ≤ |m| ≤ mˇ }, which is also the set of all payoﬀ-relevant market structures. Consider any (m, k) pair such that m ∈ M and mk > 0, wE(m, ·, k) and wS(m, ·, k) are computed in the step with indexing market structure (0,..., 0, mk, mk+1, . . . , mkˇ). For any (m, k) pair such that m ∈ M,

αS(m, ·, k) is computed in the step with indexing market structure (0,..., 0, 1, mk+1, . . . , mkˇ). For any m such that m ∈ M and m1 > 0, αE(m, k) is computed in the step with indexing market structure m. Therefore, wE, wS, αS, αE for all payoﬀ-relevant (m, c, k) are computed in Algorithm2.

Second, because all αS, αE, wE, wS required to compute the fixed point of T in each step have been either initialized or determined in previous steps, T is always a well-defined contraction mapping with a unique fixed point. Then, wE is always uniquely determined, as well as wS and αE. It remains to show that αS is also always well-defined, in particular when it is determined from equation (11). Note that for any (m, c, k) such that k = k(m) ≡ min{j; mj > 0}, when computing wE(m, c, k) using T , we always use the initialized value + + αS(m, c, k ) = 1 for all k ≥ k, which leads to the condition that

wE(m, c, k) = max{0, wS(m, c, k)}.

This implies that when wE(m, c, k) = 0, wS(m, c, k) ≤ 0. Recall that tn Procedure3, when i determining p using equation (11) in step i, it is indeed the case that wE(m, c, k ) = 0 and i i i i wS(m, c, k ) ≤ 0. Then, (i) if wS(m , c, k ) = wS(m − (mki − 1)ιki , c, k ) > 0, then the i right hand side of equation (11) changes continuously from wS(m − (mki − 1)ιki , c, k ) > 0 to wS(m, c, k) ≤ 0 when p changes from 0 to 1. This means that there exists at least one i i p ∈ [0, 1) to satisfy equation (11); (ii) if wS(m , c, k ) ≤ 0, 0 can be assigned if no p is found to satisfy equation (11). Therefore, we conclude that αS is always well-deﬁned (although it can take multiple values if multiple p solve equation (11)).

36 1 Next, we show that αS satisfies the requirement in Definition1 by proving wE(m, c, k ) ≥ 2 1 2 wE(m, c, k ) for all m, c and k ≥ k . To this end, for any computed wE, define a functional h ˇ i N βπ(ιkˇ,c,ˇ k) space G containing all functions gE : M × [ˆc, cˇ] × K → 0, 1−β such that gE ≤ wE, 1 2 1 2 and gE(m, c, k ) ≥ gE(m, c, k ) for all (m, c) and k = k + 1, with equality holds only when 1 2 N N gE(m, c, k ) = gE(m, c, k ) = 0. We aim to prove T : G → G .

Let gS denote the analogous post-exit value computed by equation (2) using gE. Under N 1 2 1 2 Assumptions1, for all gE ∈ G , gS(m, c, k ) > gS(m, c, k ) for all m, c and k = k + 1. 1 Consider the following cases when (T gE)(m, c, k ) is being computed in Algorithm2, noting 1 that by the OLS ordering of the algorithm, at this moment, αS(m, c, k ) remains at its initial 2 value 1 and αS(m, c, k ) has been determined in previous computation by Procedure3.

2 1 2 1. If αS(m, c, k ) = 1, since both type-k , k ﬁrms survive with probability one, they

expect same post-exit market structure, denoted by MS.

1 1 2 2 (T gE)(m, c, k ) = E gS(MS, c, k ) ME = m > E gS(MS, c, k ) ME = m = (T gE)(m, c, k ).

2 2 2. If 0 < αS(m, c, k ) < 1, then αS(m, c, k ) = p with p ∈ [0, 1) solving

m −1 2 k2 k −1 m 2 − 1 X m 2 −1−j j k X 2 p k (1 − p) g (m − jι 2 − m ι , c, k ) = 0. j S k i i j=0 i=1

2 2 The right hand side is nothing but (T gE)(m, c, k ). Therefore, (T gE)(m, c, k ) = 0. N Since gS ∈ G , we have

 m −1 2  k2 k −1  m 2 − 1  1 X m 2 −1−j j k X 1 (T g )(m, c, k ) = max 0, p k (1 − p) g (m − jι 2 − m ι , c, k ) > 0. E j S k i i  j=0 i=1 

2 2 2 N 2 3. If αS(m, c, k ) = 0, then gE(m, c, k ) ≤ wE(m, c, k ) = 0 for gE ∈ G . Since wE(m, c, k ) = ∞ 2 2 2 (T gE)(m, c, k ) and T is a monotone operator, 0 = wE(m, c, k ) ≥ (T gE)(m, c, k ) for N 1 2 all gE ∈ G . Thus, (T gE)(m, c, k ) ≥ 0 ≥ (T gE)(m, c, k ).

N N 1 By point-wise comparison, we conclude that T : G → G , hence wE(m, c, k ) ≥ 2 1 2 1 wE(m, c, k ) for all m, c and k = k + 1. The proof also verifies that (T gE)(m, c, k ) > 0 2 1 whenever αS(m, c, k ) > 0. Since T is a monotone operator, it means that wE(m, c, k ) = ∞ 1 1 (T gE)(m, c, k ) > 0. Given that in Algorithm2 αS is set to be 1 if and only if wE(m, c, k ) > 1 2 0, αS(m, c, k ) = 1 whenever αS(m, c, k ) > 0. So αS satisfies the requirement in Definition 1.

Finally, we prove that (αS, αE) forms an Markov-perfect equilibrium. To this end, we ﬁrst show that wE constructed by Algorithm2 is the post-entry payoﬀ under strategy ( αS, αE).

Then, we show that given wE as payoﬀ, (αS, αE) satisﬁes one-shot deviation proofness.

37 We begin with showing that wE is the post-entry payoﬀ under (αS, αE) in the ﬁrst step 1 1 1 ˇ of Algorithm2, where m =mι ˇ kˇ. In this step, HS = {(m , c, k)|c ∈ [ˆc, cˇ]}. When computing 1 1 S wE(HS), we use αS(HS) = 1 for all H1 , αE(·) = 0, and wE(·) = 0. According to equation

(1), if wE is the post-entry payoﬀ under strategy (αS, αE), then it satisﬁes

1 1 ˇ 1 ˇ wE(HS) = αS(HS)E wS(MS, c, k) ME = m ,C = c, K = k .

1 ˇ The construction of αS in Procedure3 implies that αS(m , c, k) = 1 if and only if 1 ˇ 1 ˇ 1 ˇ wE(m , c, k) > 0, and αS(m , c, k) < 1 if and only if wE(m , c, k) = 0. Also, note that ˇ 1 ˇ 1 ˇ 1 ˇ E wS(MS, c, k) ME = m ,C = c, K = k = wS(m , c, k) if αS(m , c, k) = 1. Then, the 1 above condition for wE(HS) under the constructed αS is equivalent to

1 1 1 1 wE(HS) = max{0,I(wE(HS) > 0)wS(HS)} = max{0, wS(HS)}.

1 By setting αS(HS) = 1, the right hand side of T is identical to this condition. Therefore, 1 1 setting αS(HS) = 1 is computationally equivalent to using αS(HS) determined by Procedure 1 3, i.e., both give the same wE(HS). Also, under αE(·) = 0, no firm will further enter. This 1 means that wE(HS) computed as the fixed point of T is the post-entry payoff under strategy

(αS, αE). 1 ˇ 1 ˇ Consequently, for all c, wS(m , c, k) computed by equation (2) using wE(m , ·, k) is the post-survival payoﬀ under strategy (αS, αE).

Then suppose that the 1, . . . , i−1-step of Algorithm2 have computed the wE(m, c, k) and Sj=i−1 j wS(m, c, k) for all (m, k) ∈ j=1 Mmj × {k(m )} and all c as the payoﬀs under (αS, αE). Then, Procedure3 in the ﬁrst i − 1-step computes the following part of (αS, αE) for all c,

Sj=i−1 j i • αS(m, c, k) for all (m, k) ∈ {(m, k); (m, k) ∈ j=1 Mmj ×{k(m )}, m−nιki 6= m , ∀n ∈ N}.

Sj=i−1 j • αE(m, c) for all m ∈ {m; m ∈ j=1 Mmj with k(m ) = 1}.

i Recall that k(m) ≡ min{j; mj > 0}. Now, in the i-th step of the algorithm, HS ∈ i i i {(m, c, k); m ∈ Mmi , c ∈ [ˆc, cˇ] , k = k }. To make sure that wE(HS) and wS(HS) take their values under (αS, αE), we need to use in the construction of T the strategy αS(m, ·, k) for 0 0 all m ∈ Mmi and k such that mk > 0, αE(n + jι1, ·) for all j ∈ N and all possible n , and i,0 0 0 0 0 i i wE(HS ) for (m , k ) such that m ∈/ Mmi and k 6= k , conditional on type-k ﬁrms having positive payoﬀ. We check if the required values are in place.

i 1. From the argument for step-1 computation, the initialized value αS(m, c, k ) = 1 leads i i to the same condition for wE(m, c, k ) as the αS(m, c, k ) computed by Procedure3. i So, although αS(m, c, k ) has not been obtained, we can set it to 1.

38 + + i + 2. For any (m, k ) such that m ∈ Mmi and k > k , as we have shown, αS(m, c, k ) = 1 i − conditional wE(m, c, k ) > 0, which is the same as the initialized value. For any (m, k ) − i i i such that m ∈ Mmi and k < k , note that m 6= m because mk− = 0. By the deﬁnition i i of Mmi , for all m ∈ Mmi \{m }, m m (so there is some j < i-step such that its j i i − indexing market structure m = m) and m − bιk 6= m , ∀b ∈ N. Therefore, αS(m, c, k ) for all k− < ki have been computed.

i 3. Since according to αS, all ﬁrms with type equal or better than k survive, which, 0 i 0 i together with non-regressive type evolution, implies that n m and n + bι1 m 0 0 for all n and all b ∈ N. Therefore, for |n + bι1| ≤ mˇ , there is some j < i-th step with j 0 0 indexing market structure m = n + bι1. So, these αE(n + bι1)’s values have been 0 0 computed in the j-th step by Procedure3. For any n such that |n + ι1| > mˇ , we use 0 the initialized value αE(n + ι1) = 0.

4. Based on the above argument, for any (m0, k0) following the transition governed by 0 i 0 i 0 0 i 0 0 0 0 i (αS, αE), m m and k ≥ k . If m ∈/ Mm and k 6= k , define m (k ) = (0,..., 0, mk0 , . . . , mkˇ), the market structure which has exactly the same number of type-k0 or better firms as m0 does, but no type-k0 −1 or worse firm. Then, m0(k0) mi, which means that there is j 0 0 some j < i-th step such that its indexing market structure m = (0,..., 0, mk0 , . . . , mkˇ). 0 0 0 Since m ∈ Mmj , wE(m , ·, k ) is then computed in the j-th step. So, all necessary wE’s values have been computed.

i Since all the required values of wE, αS, αE have been obtained in earlier steps, wE(HS) is i computed as the payoﬀ under (αS, αE), so as wS(HS).

Then, we verify that (αS, αE) is an equilibrium strategy corresponding to wE, wS. To this end, we show that αS(m, c, k) satisﬁes (4) for all (m, c, k), if all other ﬁrms follow αS as well. For any (m, c, k), consider the following cases

1. If wE(m, c, k) > 0, the algorithm sets αS(m, c, k) = 1. The right-hand-side of (4) is

arg max awE(m, c, k) = {1} 3 αS(m, c, k). a∈[0,1]

2. If wE(m, c, k) = 0, then the algorithm sets αS ∈ [0, 1). Since any αS computed by − Algorithm2 satisﬁes the requirement in Deﬁnition1, it is implied that αS(m, c, k ) = 0 − Pk−1 for all k < k. Hence, wE(m, c, k) = wE(m − i=1 miιi, c, k) = max{0, wS(m − Pk−1 Pk−1 i=1 miιi, c, k)} and wS(m − i=1 miιi, c, k) ≤ 0. We look at three subcases, Pk−1 (a) If wS(m − (mk − 1)ιk − i=1 miιi, c, k) > 0, the algorithm sets αS(m, c, k) = p ∈ [0, 1) to satisfy

mk−1 k−1 X mk − 1 X pmk−1−j(1 − p)j w (m − jι − m ι , c, k) = 0, j S k i i j=0 i=1

39 The right-hand-side of (4)

m −1 k m − 1 k−1 X mk−1−j j k X arg max a p (1−p) wS(m−jιk − miιi, c, k) = [0, 1] 3 αS(m, c, k). a∈[0,1] j j=0 i=1

Pk−1 (b) If wS(m−(mk −1)ιk − i=1 miιi, c, k) > 0 and αS(m, c, k) ∈ [0, 1) solves the same polynomial as above, same result holds for αS(m, c, k). Pk−1 (c) If wS(m − (mk − 1)ιk − i=1 miιi, c, k) ≤ 0 and αS(m, c, k) = 0. All other type-k ﬁrms will exit from the market, so the right-hand-side of (4) is

k−1 X arg max awS(m − (mk − 1)ιk − miιi, c, k) = {0} 3 αS(m, c, k). a∈[0,1] i=1

For any (m, c, k) such that mk = 1, consider the following cases

1. If wE(m, c, k) > 0, then the algorithm sets αS(m, c, k) = 1. The right-hand-side of (4)

is arg maxa∈[0,1] awS(m, c, k) = {1} 3 αS(m, c, k).

2. If wE(m, c, k) = 0, then αS can not be 1. From the same argument as above, wE(m, c, k) = Pk−1 Pk−1 Pk−1 wE(m− i=1 miιi, c, k) = max{0, wS(m− i=1 miιi, c, k)}. So wS(m− i=1 miιi, c, k) ≤ Pk−1 0 and the right-hand-side of (4) is arg maxa∈[0,1] awS(m − i=1 miιi, c, k) = {0} 3 αS(m, c, k).

Therefore, αS satisfies (4). To show that αE satisfies (3), first note that αE(m, c) is determined in the step with indexing market structure m, while wE(m+bι1, c, 1) is computed in step with indexing market structure m + bι1, which is (weakly) lexicographically superior than m. Therefore, wE(m + bι1, c, 1) has been determined as a post-entry payoff under αS.

Then, if all potential entrants are using αE, according to (1), post-entry payoﬀ is wE(m +

Jι1, c, 1) where J is the largest possible number such that wE(m+Jι1, c, 1)−ϕ > 0. Therefore,

αE satisﬁes (3). So, (αS, αE) is the equilibrium strategy. This completes the proof for Lemma3.

With Lemma3 in hand, we proceed to prove Propositions2 and3.

Proof for Proposition2. To prove this proposition, we again establish a lemma ﬁrst.

Lemma 4. If vE is the post-entry payoff in a payoff-monotone natural Markov-perfect equilibrium, it necessarily satisfies that vE(m, c, k) > 0 if and only if E[vS(MS, c, k)|ME = m] > 0, or

vE(m, c, k) = max{0, E[vS(MS, c, k)|ME = m]},

− − where the expectation is computed given all equilibrium values aS(m, c, k ) for all k < k, a + + tentative rule aS(m, c, k) = 1, and aS(m, c, k ) = 1 for all k > k.

40 Proof. In any symmetric equilibrium, vE(m, c, k) > 0 only if all ﬁrms with type k survive. In any natural equilibrium, this also implies that all ﬁrms with type k+ survive as well. Therefore, the “only if” part is true.

The “if” part is true because (i) if E[vS(MS, c, k)|ME = m] > 0 and aS(m, c, k − 1) > 0, then it must be the case that in natural equilibrium vE(m, c, k − 1) > 0. Also, according to + Deﬁnition1, aS(m, c, k) = 1 and aS(m, c, k ) = 1. Then, vE(m, c, k) ≥ vE(m, c, k − 1) > 0; (ii) if E[vS(MS, c, k)|ME = m] > 0 and aS(m, c, k − 1) < 0, then in a natural equilibrium − − Pk−1 aS(m, c, k ) = 0 for all k < k, and E[vS(MS, c, k)|ME = m] = vS(m − i=1 miιi, c, k) > 0. Recall that we have shown in the proof for Proposition1 that in the duopoly model, Lemma1 ensures that vE(2ιk, c, k) > 0 if and only if vS(2ιk, c, k) > 0. Applying an analogous reasoning, Pk−1 we know that in a payoﬀ-monotone equilibrium, vE(m, c, k) > 0 if vS(m − i=1 miιi, c, k) > 0.

Lemma4 gives a necessary condition for the post-entry payoff in a payoff-monotone 1 ˇ 1 natural Markov-perfect equilibrium. For vE(m , c, k), where m =mι ˇ kˇ is the indexing market structure in the first step of Algorithm2, this condition can be written as

1 ˇ 1 ˇ vE(m , c, k) = max{0, vS(m , c, k)}.

1 ˇ In the first step of Algorithm2, wE(m , c, k) is uniquely computed by the contraction mapping generated by the above condition. Thus, it is the only payoff function satisfying the necessary condition for a payoff-monotone equilibrium. Providing that such equilibrium exists, its 1 ˇ 1 ˇ 1 ˇ post-entry payoff vE(m , c, k) has a unique value wE(m , c, k) for all c. Also, vS(m , c, k) = 1 ˇ wS(m , c, k) for all c.

In any succeeding step i of Algorithm2, with αS either properly initialized or computed i in the previous steps as its equilibrium value (this is shown in Lemma3), wE(m, c, k ) is computed as the unique payoff under (αS, αE) that satisfies such necessary condition for all c and all m ∈ Mmi . Moreover, the (αS, αE) constructed in Procedure3 is also the unique equilibrium strategy given that the wE, wS computed in previous steps are unique equilibrium payoffs vE, vS. αE’s uniqueness trivially follows its construction. The uniqueness of αS is due to the monotonicity of (the previously computed part of) wS: When using (11) to compute the mixing probability i i i p, because wS(m − (mki − 1)ιki , c, k ) ≥ wS(m − (mki − 2)ιki , c, k ) ≥ ... ≥ wS(m, c, k ), the i right hand side (11) changes continuously and monotonically from wS(m−(mki −1)ιki , c, k ) > i 0 to wS(m, c, k ) ≤ 0 when p changes from 0 to 1. Therefore, there is only one p ∈ [0, 1) that satisfies (11). So, αS is single valued.

Therefore, if there exists a payoﬀ-monotone equilibrium, (αS, αE) forms the unique equilibrium and wE and wS are the unique equilibrium payoﬀs. The equilibrium is subsequently unique.

41 Proof for Proposition3. First, note that from the definition of a renegotiation-proof natural Markov-perfect equilibrium, all firms with a same type survive for sure if and only if joint continuation gives them positive post-survival payoff. This implies that (i) any such equilibrium’s post-entry equilibrium payoff must satisfy the condition in Lemma4; (ii) if any natural Markov-perfect equilibrium’s post-entry payoff satisfies the condition in Lemma4, such equilibrium is renegotiation-proof.

Since we have shown in Lemma3 that Algorithm2 always gives some ( αS, αE) to form a natural Markov-perfect equilibrium. We have also shown in the proof for Proposition2 that wE satisﬁes the necessary condition in Lemma4. Therefore, ( αS, αE) forms a renegotiation- proof natural Markov-perfect equilibrium.

C An Example of Multiple Equilibria

We construct a three-firm two-type example where the equilibrium payoff is not weakly decreasing in the number of same-type competitors (m ˇ = 3 (by setting π(n, c, k) < 0 for any (n, c, k) if n has more than 3 firms) and kˇ = 2). −6 Consider the following sequence of ct: c1 = 1, c2 = 1e , ct = 5, for all t ≥ 3. The number of consumers drops to nearly zero in the second period but is boosted to a high level in the third period, and stays high afterwards. We set β = 0.5, κ(L) = κ(H) = 4, ϕ = 1, and

ΠLH = ΠLL = 0.5. We specify π as π(n, c, k) = cπ(n, k) − κ. Some parts of the per-consumer producer surplus π are summarized in the following table:

π(·, L)/π(·, H) 1ιH 2ιH 3ιH

+0ιL /102 /100 /1

+1ιL 99/101 0.89/1.24

+2ιL 1.23/1.25 One feature of this surplus structure is that a duopoly market promises much higher per consumer surplus than a triopoly market does. The duopoly-triopoly surplus difference over- whelmingly dominates the L, H-type difference in surplus. Since from period 3 onwards, the model is essentially an infinitely repeated game, we can use backward induction to compute 13 the equilibrium payoffs . Unlike the results stated in Lemma1, vS is not always monotonic in the number of firms,

vS(ιH, c1, H) = −1.1821, vS(2ιH, c1, H) = 246, vS(3ιH, c1, H) = −1.5

Not surprisingly, the low triopoly surplus implies a low payoff if continuing as one of three type-H firms. What is counter-intuitive is that the payoff to continuing as a duopolist is

13We provide the details of such computation in an online appendix.

42 better than the payoff to continuing as a monopolist under c1. This is because in period 2, under a low c2, a duopoly firm and a monopoly firm make similar flow profit and similar large losses. However, duopoly firms can, by jointly remaining active, preempt any further entrants and enjoy a high duopoly profit after demand increases to a high level in period 3.

The future duopoly payoff compensates the loss in period 2, and make vS(2ιH, c1, H) positive. In contrast, a monopoly market attracts two entrants for sure in period 2, which results in a triopoly market from period 3 onwards. Because demand increases to a high level in period 3, none of these firms will exit and, given the per consumer surplus structure, they will all earn a substantially lower payoff than duopoly firms, which can not compensate the loss in period 2. Consequently, vS(ιH, c1, H) is negative.

Given the computed non-monotone equilibrium payoff, (aS, aE) with aS(2ιH, c1, H) = 1 is still an natural equilibrium. However, if one duopoly firm chooses to exit with probability 1, the rival firm receives -1.1821 if continuing alone and hence will choose to exit with probability −1.1821 −3 1 as well. Similarly, if one firm chooses to survive with probability −1.1821−246 = 4.782e , the other firm is indifferent between exiting and survival. Therefore, two other natural equilibria −3 with aS(2ιH, c1, H) = 0 and aS(2ιH, c1, H) = 4.782e exist. −3 Note that when both firms choose aS(2ιH, c1, H) = 0 or aS(2ιH, c1, H) = 4.782e , they receive zero payoffs. By ”renegotiating” on jointly choosing aS(2ιH, c1, H) = 1, they can strictly improve their equilibrium payoffs. Henceforth, we only restrict attention to equilibria in which there is no room for this type of one-shot joint improvement. Unfortunately, this type of equilibria may not be unique. Since joint continuation and continuing as monopolist both render payoffs negative, in a one-shot renegotiation-proof equilibrium, a triopoly firm either chooses aS(3ιH, c1, H) = 0 or set aS(2ιH, c1, H) = p, where p solves

2 p2v¯(3ι , c , H) + p(1 − p)¯v(2ι , c , H) + (1 − p)2v¯(ι , c , H) = 0. H 1 1 H 1 H 1

This quadratic equation has two roots, p1 = 0.0024 and p2 = 0.997, both between 0 and

1. Hence, there are three one-shot renegotiation-proof equilibria with aS(3ιH, c1, H) equal to

0, p1 and p2, respectively.

43 Online Appendix for Simple Markov-Perfect Industry Dynamics

Jaap H. Abbring∗ Jeﬀrey R. Campbell† Nan Yang‡

November 30, 2010

Contents

I Details of Proofs1 I.1 Details for the Proof of Lemma1...... 1 I.2 Details for the Proof of Proposition1...... 9

II Computational Techniques 10 II.1 Computing A Firm’s Beliefs about Next Period’s State...... 10 II.2 Constructing the Type Transition Matrices in Matlab...... 11 II.2.1 The Problem...... 11 II.2.2 The Dimensionality...... 12 II.2.3 Recursive Construction of Πm ...... 12 II.2.4 The typetransition.m Function...... 21 II.3 Computing All Renegotiation-proof Natural Markov-Perfect Equilibria... 21

III Computational Details 23 III.1 Pencil-and-Paper Computation behind Figure3...... 23 III.2 Computing the Example of Multiple Equilibria in Section 4.2...... 30

∗CentER, Department of Econometrics & OR, Tilburg University. E-mail: [email protected] †Federal Reserve Bank of Chicago and NBER. E-mail: [email protected] ‡VU University Amsterdam and Tinbergen Institute. E-mail: [email protected] JEL Code: L13 I Details of Proofs

I.1 Details for the Proof of Lemma1

Proof. We prove that, in any natural Markov-Perfect Equilibrium, vE satisfies the following properties for all k. ˇ Property 1. For a given k ∈ K, and for all x such that 0 ≤ x ≤ k, if vE(ιk + ιx, ·, k) is weakly decreasing in x, we say that vE satisfies Property1 for k. ˇ Property 2. For a given k ∈ K, and for all x such that 0 ≤ x ≤ k, if vE(ιk + ιx, ·, x) is weakly increasing in x, we say that vE satisfies Property2 for k.

Property 3. For a given k ∈ K, and for all x such that 1 ≤ x ≤ k − 1, if vE(ιk + ιx, ·, x) ≤ vE(ιk−1 + ιx, ·, x), we say that vE satisﬁes Property3 for k.

Property 4. For a given k ∈ K, if vE(ιk, ·, k) ≥ vE(ιk−1, ·, k − 1), we say that vE satisﬁes Property4 for k. First, we prove these properties for k = kˇ. Then, for any k < kˇ, suppose that those properties hold good for k + 1, k + 2,..., kˇ, we prove that they also hold for k. In this way, we prove that these properties hold for all k ∈ K. Eventually, we proof Lemma1 using Property1.

Now suppose (aS, aE) forms a natural equilibrium with equilibrium payoﬀ vS, vE. Deﬁne F to be the space of all functions   kˇ ˇ  X  βπ(ιkˇ, c,ˇ k) f : (n , . . . , nˇ): n ≤ 2 × [ˆc, cˇ] × → 0, , E 1 k i K 1 − β  i=1  and T a : F → F with For any k1, k2 ∈ {0, 1,..., kˇ}, if k1 ≥ k2,

( kˇ kˇ a 1 2 X X ˜ (T fE)(ιk1 + ιk2 , c, k ) = max 0, aS(ιk1 + ιk2 , c, k ) Πk1,iΠk2,jfE(ιi + ιj, c, i) i=k1 j=k2 kˇ ) 2 X ˜ +(1 − aS(ιk1 + ιk2 , c, k )) Πk1,ifE(ιi, c, i) i=k1 2 ˜ 10 2 2 1 1 = max 0, aS(ιk1 + ιk2 , c, k )E[E[fE(ιK10 + ιK20 , c, K )|K = k ]|K = k ] 2 ˜ 10 1 1 +(1 − aS(ιk1 + ιk2 , c, k ))E[fE(ιK10 , c, K )|K = k ] . (1)

1 If k1 < k2,

 ˇ ˇ   k k  a 1 X X ˜ (T fE)(ιk1 + ιk2 , c, k ) = max 0, Πk1,iΠk2,jfE(ιi + ιj, c, i)  i=k1 j=k2  n ˜ 10 2 2 1 1 o = max 0, E[E[fE(ιK10 + ιK20 , c, K )|K = k ]|K = k ] .(2)

˜ 1 With fE(ιk1 + ιk2 , c, k ) denoting the post-type-transition payoff associate with post-entry 1 payoff fE when current demand is c, the firm of interest has progressed to type-k , and its rival has progressed to type-k2.

˜ 1 0 1 0 0 fE(ιk1 + ιk2 , c, k ) ≡ βE[π(ιk1 + ιk2 ,C , k ) + aE(ιk1 + ιk2 + ι1,C )fE(ιk1 + ιk2 + ι1,C , i) 0 0 + [1 − aE(ιk1 + ιk2 + ι1,C )] fE(ιk1 + ιk2 ,C , i)|C = c]

1 2 where, for definiteness, aE(ιk1 + ιk2 + ι1, c) ≡ 0 if k , k > 0. The space F is a Banach space (complete with the supremum norm). T a satisfies Black- well’s sufficient properties for a contraction mapping. The equilibrium payoff vE is the unique a fixed point of T . We prove that vE satisfies Properties1–4 for all k by showing that the fixed point of T a lies in the space in which all functions satisfy these properties. To this end, we gradually narrow down the space that this fixed point is in, to eventually reach the space of all functions that satisfy Properties1–4 for all k. 0 First, denote a subspace of F in which any functions fE satisfy fE ≥ vE as F . Because 0 0 a 0 0 F is also a non-empty Banach space and vE ∈ F , T : F → F . We henceforth focus on F 0. Before proceeding to verify the properties, we introduce a Lemma which we will use repeatedly. Recall that firm types’ evolution has the first-order stochastic dominance property, as stated in Assumption3. The following Lemma exploits this property Lemma 1. X, Y are random variables with densities F and G respectively. If X first-order stochastically dominates Y , then E[h(X)] ≥ E[h(Y )] for all weakly increasing function h.

Prove Property2 for k = kˇ. We take two steps to prove Property2 for k = kˇ.

1 0 a ˇ (i). Consider Fkˇ , a subspace of F in which any function fE satisﬁes (T fE)(2ιkˇ, ·, k) = ˇ ˇ ˇ ˇ vE(2ιkˇ, ·, k) and fE(2ιkˇ, ·, k) ≤ fE(ιkˇ + ιx, ·, k) for 0 ≤ x ≤ k. This is also a complete ∗ ∗ ˇ ˇ metric space. Since at least the function fE which satisﬁes fE(2ιkˇ, ·, k) = vE(2ιkˇ, ·, k) ∗ ˇ ∗ ˇ ˇ 1 and fE(2ιkˇ, ·, k) = fE(ιkˇ + ιx, ·, k) for 0 ≤ x ≤ k is in Fkˇ , this subspace is nonempty.

2 a 1 1 a We aim to prove that T : Fkˇ → Fkˇ . Since vE is the unique fixed point of T , this 1 ˇ ˇ result ensures that vE ∈ Fkˇ and thus vE(2ιkˇ, ·, k) ≤ vE(ιkˇ, ·, k). In a symmetric equilibrium, any rival’s exit implies that both firms expect non-positive payoffs from continuing. Therefore, firms earn positive expected payoffs only when both firms continue with probability 1. So

ˇ 0 ˇ 0 ˇ ˜ ˇ vE(2ιkˇ, c, k) ≤ max{0, βE[π(2ιkˇ,C , k)+vE(2ιkˇ,C , k)|C = c]} = max{0, fE(2ιkˇ, c, k)}.

1 ˜ ˇ ˜ ˇ Note that for all fE ∈ Fkˇ , we have for any (aS, aE), fE(ιkˇ + ιx, ·, k) ≥ fE(2ιkˇ, ·, k) for ˇ ˜ ˇ ˜ ˇ ˜ ˇ 0 ≤ x ≤ k. Therefore, E[fE(ιkˇ + ιX0 , ·, k)|X = x] ≥ fE(2ιkˇ, ·, k), and fE(ιkˇ, ·, k) ≥ ˜ ˇ ˇ fE(2ιkˇ, ·, k). Then, for any 0 ≤ x ≤ k, we have

a ˇ (T fE)(ιkˇ + ιx, c, k) n ˜ ˇ ˜ ˇ o = max 0, aS(ιkˇ + ιx, c, x)E[fE(ιkˇ + ιX0 , c, k)|X = x] + (1 − aS(ιkˇ + ιx, c, x))fE(ιkˇ, c, k) ˜ ˇ ≥ max{0, fE(2ιkˇ, c, k)} ˇ ≥ vE(2ιkˇ, c, k) a ˇ = (T fE)(2ιkˇ, c, k)

a 1 1 ˇ ˇ ˇ Therefore, T : Fkˇ → Fkˇ and vE(2ιkˇ, ·, k) ≤ vE(ιkˇ + ιx, ·, k) for 0 ≤ x ≤ k. This further ˇ ˇ ensures that vS(2ιkˇ, ·, k) ≤ vS(ιkˇ, ·, k) for any aS, aE. Then, an analogous argument to the one on the simultaneous-move survival game in Section ?? leads to

ˇ 0 ˇ 0 ˇ vE(2ιkˇ, c, k) = max 0, βE π(2ιkˇ,C , k) + vE 2ιkˇ,C , k C = c . (3)

The right-hand-side of (3) deﬁnes a contraction mapping with a unique ﬁxed point ˇ vE(2ιkˇ, ·, k). So

a ˇ ˜ ˇ (T fE)(2ιkˇ, c, k) = max{0, fE(2ιkˇ, c, k)}.

1 2 (ii). We move on to a subspace of Fkˇ , which we denote by Fkˇ . In this subspace, any function ˇ fE satisﬁes that fE(ιkˇ + ιx, ·, x) is weakly increasing in x for 0 ≤ x ≤ k. We will further a 2 2 2 ˜ show that T : Fkˇ → Fkˇ . Note that for fE ∈ Fkˇ , fE(ιkˇ + ιx, ·, x) is weakly increasing in x as well. For any k1, k2 such that 1 ≤ k1 ≤ k2 ≤ kˇ, we use K10 ,K20 to denote the random variables for the types succeeding K1 = k1,K2 = k2 respectively. K20 10 ˜ stochastically dominates K . So, according to Lemma1, fE shares the monotonicity

3 2 10 1 1 20 2 2 ˜ 0 ˜ 0 property as fE in Fkˇ , E[fE(ιkˇ + ιK1 , ·,K )|K = k ] ≤ E[fE(ιkˇ + ιK2 , ·,K )|K = k ]. Therefore,

a 1 n ˜ 10 1 1 o (T fE)(ιkˇ + ιk1 , c, k ) = max 0, E[fE(ιkˇ + ιK10 , c, K )|K = k ] n ˜ 20 2 2 o ≤ max 0, E[fE(ιkˇ + ιK20 , c, K )|K = k ] a 2 = (T fE)(ιkˇ + ιk2 , c, k )

a 2 2 This result guarantees that T : Fkˇ → Fkˇ . Therefore, vE(ιkˇ + ιx, ·, x) is weakly increasing in x and Property2 is satisﬁed for k = kˇ. Because in natural equilibrium,

0 0 0 0 vS(ιkˇ + ιx, c, x) = βE [π(ιkˇ + ιX0 ,C ,X ) + vE(ιkˇ + ιX0 ,C ,X ) C = c, X = x] .

together with Assumption1, it is also true that vS(ιkˇ + ιx, ·, x) is weakly increasing in

x, and so is the survival rule aS(ιkˇ + ιx, ·, x).

ˇ 2 3 Prove Property1 for k = k. Next, we focus on a subspace of Fkˇ , denoted by Fkˇ . In this ˇ subspace any function fE must satisfy that fE(ιkˇ + ιx, ·, k) is weakly decreasing in x for any ˇ 3 ˜ ˇ 0 ≤ x ≤ k. Note that for fE ∈ Fkˇ , fE(ιkˇ +ιx, ·, k) is weakly decreasing in x as well. Therefore, 1 2 ˇ ˜ ˇ 2 2 ˜ ˇ 1 1 for any 0 ≤ k ≤ k ≤ k, E[fE(ιkˇ + ιK20 , ·, k)|K = k ] ≤ E[fE(ιkˇ + ιK10 , ·, k)|K = k ]. Then, using the monotonicity of survival rule that we derived above, a ˇ (T fE)(ιkˇ + ιk2 , c, k) n 2 ˜ ˇ 2 2 2 ˜ ˇ o = max 0, aS(ιkˇ + ιk2 , c, k )E[fE(ιkˇ + ιK20 , c, k)|K = k ] + (1 − aS(ιkˇ + ιk2 , c, k ))fE(ιkˇ, c, k) n 1 ˜ ˇ 2 2 1 ˜ ˇ o ≤ max 0, aS(ιkˇ + ιk1 , c, k )E[fE(ιkˇ + ιK20 , c, k)|K = k ] + (1 − aS(ιkˇ + ιk1 , c, k ))fE(ιkˇ, c, k) n 1 ˜ ˇ 1 1 1 ˜ ˇ o ≤ max 0, aS(ιkˇ + ιk1 , c, k )E[fE(ιkˇ + ιK10 , c, k)|K = k ] + (1 − aS(ιkˇ + ιk1 , c, k ))fE(ιkˇ, c, k) a ˇ = (T fE)(ιkˇ + ιk1 , c, k) a 3 3 ˇ Therefore, T : Fkˇ → Fkˇ and vE(ιkˇ + ιx, ·, k) is weakly decreasing in x, so Property1 is satisﬁed for k = kˇ.

Prove Property3 for k = kˇ. Before proving this property, we need to prove Properties2 and1 for k = kˇ − 1. We skip the details because later we will demonstrate how to do so for ˇ 3 any k. Now suppose that we have veriﬁed these two properties for k = k − 1, then vE ∈ Fkˇ−1 3 3 ˇ ˇ where Fkˇ−1 is deﬁned analogously to Fkˇ . Hence, for 0 ≤ x ≤ k, vE(ιkˇ−1 + ιx, ·, k − 1) is weakly decreasing in x and vE(ιkˇ−1 + ιx, ·, x) is weakly increasing in x. Then, denote the

4 3 subspace of Fkˇ−1 in which any fE functions such that fE(ιkˇ + ιx, ·, x) ≤ fE(ιkˇ−1 + ιx, ·, x) for ˇ 4 all x such that 0 ≤ x ≤ k − 1 by Fkˇ−1. ˜ ˜ 4 Then, fE(ιkˇ + ιx, ·, x) ≤ fE(ιkˇ−1 + ιx, ·, x) holds for fE ∈ Fkˇ−1, by Lemma1, we have a n ˜ 0 o (T fE)(ιkˇ + ιx, c, x) = max 0, E[fE(ιkˇ + ιX0 , c, X )|X = x] n ˜ 0 ˇ o ≤ max 0, E[E[fE(ιK0 + ιX0 , c, X )|X = x]|K = k − 1] a = (T fE)(ιkˇ−1 + ιx, c, x)

a 4 4 Therefore, T : Fkˇ−1 → Fkˇ−1 and vE(ιkˇ + ιx, ·, x) ≤ vE(ιkˇ−1 + ιx, ·, x) for all x such that ˇ 0 ≤ x ≤ k − 1. In particular, this result ensures that aE(ιkˇ + ι1, ·) ≤ aE(ιkˇ−1 + ι1, ·).

ˇ 4 Prove Property4 for k = k. Define a subspace of Fkˇ−1 in which any function fE satisfies ˇ ˇ 5 fE(ιkˇ−1, ·, k − 1) ≤ fE(ιkˇ, ·, k), as Fkˇ−1. Because aE(ιkˇ + ι1, ·) ≤ aE(ιkˇ−1 + ι1, ·), we have ˜ ˇ ˜ ˇ a ˇ a ˇ fE(ιkˇ−1, ·, k − 1) ≤ fE(ιkˇ, ·, k) and then (T fE)(ιkˇ−1, ·, k − 1) ≤ (T fE)(ιkˇ, ·, k). Therefore, a 5 5 ˇ ˇ T : Fkˇ−1 → Fkˇ−1 and hence vE(ιkˇ, ·, k) ≥ vE(ιkˇ−1, ·, k − 1). Now suppose that for any k ≤ kˇ, we have established the following results ˇ Result 1. vE satisfies Property1 for k + 1, k + 2,..., k. That is, vE(ιk+1 + ιx, ·, x), vE(ιk+2 + ˇ ιx, ·, x), . . . , vE(ιkˇ + ιx, ·, x) are all weakly increasing in x, 0 ≤ x ≤ k. ˇ Result 2. vE satisfies Property2 for k +1, k +2,..., k. That is, vE(ιx +ιk+1, ·, k +1), vE(ιx + ˇ ˇ ιk+2, ·, k + 2), . . . , vE(ιx + ιkˇ, ·, k) are all weakly decreasing in x, 0 ≤ x ≤ k. ˇ Result 3. vE satisfies Property3 for k + 1, k + 2,..., k. That is, vE(ιk+1 + ιx, ·, x) ≥ vE(ιk+2 + ιx, ·, x) ≥ ... ≥ vE(ιkˇ + ιx, ·, x) for all 1 ≤ x ≤ k. ˇ Result 4. vE satisfies Property4 for k + 1, k + 2,..., k. That is, vE(ιk+1, ·, k + 1) ≤ ˇ vE(ιk+2, ·, k + 2) ≤ ... ≤ vE(ιkˇ, ·, k).

We then need to prove that vE satisﬁes Properties1-4 for k.

Prove Property2 for k. We follow three steps to achieve this end.

1 5 a (i). First, consider Fk , the subspace of Fk+1 in which any function f satisﬁes that (T fE)(2ιk, ·, k) = ˇ vE(2ιk, ·, k), fE(ιx + ιk, ·, k) is weakly decreasing in x, k ≤ x ≤ k, and fE(2ιk, ·, k) ≤ ∗ ∗ fE(ιk + ιx, ·, k), for all 0 ≤ x ≤ k. Note that at least a function fE with fE(2ιk, ·, k) = ∗ 1 1 1 vE(2ιk, ·, k) = fE(ιk + ιx, ·, k) for all x is in Fk , so Fk is nonempty. For any fE ∈ Fk , ˜ fE shares the properties with fE and hence also has the properties stated in Results

5 a ˇ 1–4. To prove that (T fE)(ιx + ιk, ·, k) is weakly decreasing in x, k ≤ x ≤ k, consider the following cases for any k1, k2 such that k ≤ k1 < k2 ≤ kˇ.

1 2 ˜ 0 1 1 (a) If k < k < k , according to Lemma1, E[E[fE(ιK10 +ιK0 , ·,K )|K = k]|K = k ] ≥ ˜ 0 2 2 a E[E[fE(ιK20 + ιK0 , ·,K )|K = k]|K = k ]. Thus, from equation (2), (T fE)(ιk2 + a ιk, c, k) ≤ (T fE)(ιk1 + ιk, c, k). (b) If k = k1 < k2, ﬁrst observe that

a n ˜ 0 2 2 o (T fE)(ιk2 + ιk, c, k) = max 0, E[E[fE(ιK20 + ιK0 , c, K )|K = k]|K = k ] n ˜ 0 1 o ≤ max 0, E[E[fE(ιK10 + ιK0 , c, K )|K = k]|K = k]

1 ˜ ˜ ˜ Then, because (i) for all fE ∈ Fk , fE(2ιk, ·, k) ≤ fE(ιk, ·, k) and fE(2ιk, ·, k) ≤ ˜ ˜ 0 fE(ιk + ι1, ·, k), and (ii) Result2, we obtain that E[fE(ιk + ιK0 , ·,K )|K = k] ≤ ˜ 0 ˜ 0 1 E[fE(ιK0 , ·,K )|K = k], and further E[E[fE(ιK10 + ιK0 , ·,K )|K = k]|K = k] ≤ ˜ 0 E[fE(ιK0 , ·,K )|K = k]. Then, a ˜ 0 1 1 (T fE)(ιk2 + ιk, c, k) ≤ max{0, aS(2ιk, c, k)E[E[fE(ιK10 + ιK0 , c, K )|K = k]|K = k = k] ˜ 0 +(1 − aS(2ιk, c, k))E[fE(ιK0 , c, K )|K = k]} a = (T fE)(ιk1 + ιk, c, k)

a a To prove that (T fE)(2ιk, ·, k) ≤ (T fE)(ιk + ιx, ·, k), for all 0 ≤ x ≤ k, note that,

a (T fE)(2ιk, c, k) = vE(2ιk, c, k)

≤ max {0, vS(2ιk, c, k)} n ˜ 0 1 o ≤ max 0, E[E[fE(ιK10 + ιK0 , c, K )|K = k]|K = k] ˜ 0 ≤ max{0, aS(ιk + ιx, c, x)E[E[fE(ιX0 + ιK0 , c, K )|K = k]|X = x] ˜ 0 +(1 − aS(ιk + ιx, c, x))E[fE(ιK0 , c, K )|K = k]} a = (T fE)(ιk + ιx, c, k)

The first inequality is due to equilibrium symmetric; for two type-k firm, either firm’s equilibrium payoff is bounded by payoff from joint continuation. The second inequality 0 a 1 1 is because fE ∈ F so fE ≥ vE. These results show that T : Fk → Fk . An familiar argument on the simultaneous-move survival game leads to

vE(2ιk, c, k) = max {0, vS(2ιk, c, k)} . (4)

6 The right-hand-side of (4) deﬁnes a contraction mapping with a unique ﬁxed point

v(2ιk, ·, k),

a n ˜ 0 1 o (T fE)(2ιk, c, k) = max 0, E[E[fE(ιK10 + ιK0 , c, K )|K = k]|K = k] .

1 2 (ii). We move on to a subspace of Fk , which we denote by Fk . Any function fE in this 2 subspace satisfy that fE(ιk +ιx, ·, x) is weakly increasing with x. Note that for fE ∈ Fk , ˜ fE(ιk + ιx, ·, x) is weakly increasing in x as well. Combine it with Result1, and then ˜ ˇ we have that fE(ιd + ιx, ·, x) is weakly increasing in x for all d such that k ≤ d ≤ k. ˜ 1 2 Therefore, E[fE(ιK0 +ιx, ·, x)|K = k] is weakly increasing in x. For any k , k such that 1 ≤ k1 ≤ k2 ≤ kˇ, according to Lemma1,

˜ 10 1 1 ˜ 20 2 2 [E[fE(ιK0 + ιK10 , ·,K )|K = k]|K = k ] ≤ [E[fE(ιK0 + ιK20 , ·,K )|K = k]|K = k ]. (5)

We then consider the following cases.

(a) For k1 ≤ k2 ≤ k, from equation (2), we can observe that equation (5) leads to a 1 a 2 (T fE)(ιk + ιk1 , c, k ) ≤ (T fE)(ιkˇ + ιk2 , c, k ). 1 2 ˇ (b) For k < k ≤ k ≤ k, from Result3, we have aS(ιk + ιk1 , ·, k) ≥ aS(ιk + ιk2 , ·, k). Also, from Result2 and Lemma1,

˜ i0 i i ˜ i0 i i [E[fE(ιK0 + ιKi0 , ·,K )|K = k]|K = k ] ≤ E[fE(ιKi0 , ·,K )|K = k ], i = 1, 2.

From Result4 and Lemma1,

˜ 20 2 2 ˜ 10 1 1 E[fE(ιK20 , ·,K )|K = k ] ≤ E[fE(ιK10 , ·,K )|K = k ].

a 1 a 2 Using equation (1), we can show (T fE)(ιk + ιk1 , c, k ) ≤ (T fE)(ιk + ιk2 , c, k ) by exploiting the inequalities. 1 2 ˇ a 1 a (c) For k ≤ k ≤ k ≤ k, similarly we can show (T fE)(ιk + ιk1 , c, k ) ≤ (T fE)(ιk + 2 ιk2 , c, k ) with the above results.

a 2 3 This result guarantees that T : Fk → Fk . Therefore, any equilibrium payoﬀ must ˇ satisfy that vE(ιk + ιx, ·, x) is weakly increasing in x for all 1 ≤ x ≤ k, which leads to

the same monotonicity for vS and the equilibrium survival rule.

7 3 3 Prove Property1 for k. Next, we focus on a subspace of Fk , denoted by Fk . In this subspace, any function fE satisﬁes that fE(ιx + ιk, ·, k) is weakly decreasing in x, 0 ≤ x ≤ k. 3 ˜ Note that for fE ∈ Fk , fE(ιx + ιk, ·, k) is also weakly decreasing in x, 0 ≤ x ≤ k. Combine it ˜ ˇ with Result2, and then we have that fE(ιx +ιd, ·, d) is weakly decreasing in x, 0 ≤ x ≤ k and ˇ ˜ 0 for all d such that k ≤ d ≤ k. Therefore, E[fE(ιK0 + ιx, ·,K )|K = k] is weakly decreasing in x. For any k1, k2 such that 0 ≤ k1 ≤ k2 ≤ kˇ, Lemma1 implies that

˜ 0 1 1 ˜ 0 2 2 [E[fE(ιK0 + ιK10 , ·,K )|K = k]|K = k ] ≥ [E[fE(ιK0 + ιK20 , ·,K )|K = k]|K = k ].

1 2 a Also, we have aS(ιk +ιk1 , ·, k ) ≤ aS(ιk +ιk2 , ·, k ). Therefore, it must be true that (T fE)(ιk + a a 3 3 ιk2 , c, k) ≤ (T fE)(ιk + ιk1 , c, k). So T : Fk → Fk and the equilibrium payoﬀ vE(ιx + ιk, ·, k) is weakly decreasing in x, 0 ≤ x ≤ kˇ.

3 4 Prove Property3 for k. Next, we further look into a subspace of Fk , denoted by Fk , in which any function fE satisﬁes that fE(ιk + ιx, ·, x) ≤ fE(ιk+1 + ιx, ·, x) for all x < k. Note that Result2 and Property1 ensure that fE(ιk + ιx, ·, x) ≤ fE(ιk+1 + ιx, ·, x) for all x ≥ k, 3 ˜ ˜ so for any fE ∈ Fk , fE(ιk+1 + ιx, ·, x) ≤ fE(ιk + ιx, ·, x) for all x. Combine it with Result ˜ 0 1 1 ˇ 3, and then we have E[fE(ιk1 + ιX0 , ·,X )|X = x] is weakly decreasing in k for k ≤ k ≤ k. According to Lemma1,

˜ 0 1 1 ˜ 0 E[E[fE(ιK10 + ιX0 , ·,X )|X = x]|K = k ] ≤ E[E[fE(ιK0 + ιX0 , ·,X )|X = x]|K = k].

a a Then using equation (2) we can show that (T fE)(ιk1 +ιx, c, x) ≤ (T fE)(ιk +ιx, c, x) for any a 4 4 1 x < k. Therefore, T : Fk → Fk and vE(ιk1 + ιx, c, x) ≤ vE(ιk + ιx, c, x) for all x and all k 1 ˇ such that k ≤ k ≤ k. In particular, this result ensures that aE(ιk+1 + ι1, ·) ≤ aE(ιk + ι1, ·).

4 Prove Property4 for k. Finally, define a subspace of Fk , in which any function fE 5 satisfies fE(ιk, ·, k) ≤ fE(ιk+1, ·, k + 1), as Fk . Because aE(ιk+1 + ι1, ·) ≤ aE(ιk + ι1, ·), we ˜ ˜ have fE(ιk, ·, k) ≤ fE(ιk+1, ·, k + 1) as well. Combine it with Result4, and then we have ˜ 1 1 1 ˇ that fE(ιk1 , ·, k ) is weakly increasing in k for k ≤ k ≤ k. Using Lemma1, we have a a a 5 5 (T fE)(ιk, c, k) ≤ (T fE)(ιk+1, c, k + 1) so T : Fk → Fk and Property4 is verified for k. This completes the verification for the sufficient properties for any arbitrary k. Since

(aS, aE) is also arbitrarily chosen, any natural equilibrium payoﬀ function vE must satisfy Properties1,2, and4. Then we can prove Lemma1.

8 Prove Lemma1. For any strategy (aS, aE), as a special case of Property1, vE(2ιk, c, k) ≤ ˇ vE(ιk, c, k) for any k ≤ k. To prove vS(2ιk, c, k) ≤ vS(ιk, c, k), note that

a 0 1 vS(2ιk, c, k) = E[E[˜vE(ιK0 + ιK10 , c, K )|K = k]|K = k]

a 0 vS(ιk, c, k) = E[˜vE(ιK0 , c, K )|K = k], a ˜ 1 2 wherev ˜E is deﬁned analogously as fS. For any k , k ≥ k, Property1 ensures that vE(ιk1 + 1 1 1 a 1 a 1 ιk2 , c, k ) ≤ vE(ιk1 + ι1, c, k ) ≤ vE(ιk1 , c, k ), andv ˜E(ιk1 + ιk2 , c, k ) ≤ v˜E(ιk1 + ι1, c, k ) ≤ a 1 0 1 0 v˜E(ιk1 , c, k ). Therefore E[E[vS(ιK0 + ιK10 , c, K )|K = k]|K = k] ≤ E[vS(ιK0 , c, K )|K].

I.2 Details for the Proof of Proposition1

1 2 Proof. We provide here the details to prove wE(ιk1 + ιk2 , ·, k ) ≥ wE(ιk1 + ιk2 , ·, k ). Deﬁne F to be the space of all functions   kˇ ˇ  X  βπ(ιkˇ, c,ˇ k) f : (n , . . . , nˇ): n ≤ 2 × [ˆc, cˇ] × → 0, , E 1 k i K 1 − β  i=1  and T α : F → F with

( 2 1 2 ) α 1 (Tk1,k2 f)(C), f(C) ≡ g(ιk1 + ιk2 , c, k ) if k ≤ k (T f )(ι 1 + ι 2 , c, k ) = . E k k 2 2 2 1 2 (Tk1 f)(C, k ), f(C, k ) ≡ g(ιk1 + ιk2 , c, k ) if k > k

α Thus, T is exactly assembled by Tk1,k2 and Tk1 in Algorithm1, and wE computed by Algorithm1 is the unique ﬁxed point of T α. Now consider a subspace of F, which we N 1 denote as F . In this space, any function fE satisﬁes that fE ≤ wE, fE(ιk1 + ιk2 , ·, k ) ≥ 2 1 2 fE(ιk1 + ιk2 , ·, k ) for all k > k . α N N N ˜ N We aim to prove T : F → F . For all fE ∈ F , fE ∈ F as well. Consider the following cases

2 (i). If Algorithm1 computes αS(ιk1 + ιk2 , c, k ) = 1, then Algorithm1 also prescribes 1 αS(ιk1 + ιk2 , c, k ) = 1. Substitute these survival rules into Equation (1) and use α 1 α 2 Lemma1 again, we obtain that ( T fE)(ιk1 + ιk2 , c, k ) ≥ (T fE)(ιk1 + ιk2 , c, k ).

2 (ii). If Algorithm1 computes αS(ιk1 + ιk2 , c, k ) = 0, then it must be the case that wE(ιk1 + 2 2 2 ιk2 , c, k ) = 0. For any fE ∈ F, fE(ιk1 + ιk2 , c, k ) ≤ wE(ιk1 + ιk2 , c, k ) = 0. 2 α∞ 2 α Since wE(ιk1 + ιk2 , c, k ) = (T fE)(ιk1 + ιk2 , c, k ) and T is a monotone operator, α 2 2 α 1 (T fE)(ιk1 + ιk2 , c, k ) ≤ wE(ιk1 + ιk2 , c, k ) = 0. Thus, (T fE)(ιk1 + ιk2 , c, k ) ≥ 0 ≥ α 2 (T fE)(ιk1 + ιk2 , c, k ).

9 α N N 1 By point-wise comparison, we conclude that T : F → F hence wE(ιk1 + ιk2 , ·, k ) ≥ 2 1 2 2 wE(ιk1 + ιk2 , ·, k ) for all k > k . This means that whenever wE(ιk1 + ιk2 , ·, k ) > 0, wE(ιk1 + 1 ιk2 , c, k ) > 0 as well.

II Computational Techniques

II.1 Computing A Firm’s Beliefs about Next Period’s State

The computation of the expectation in (1) requires the distribution of MS conditional on ME, given that the firm of interest survives and that all other firms use the common strategy aS. Denote the density (with respect to the appropriate dominating measure) of this distribution P i P P with pmS (·|ME = mE). Decompose mS = i mS ≡ i mS,iιi and mE = i mE,iιi, with mS,i the number of firms who have type-i in the current period and continue to next period, and mE,i the number of firms who are active when the current period’s continuation decisions i kˇ are made. Then, because MS,. . . , MS are independent conditional on ME and given that the

firm of interest survives, pmS (·|ME = mE) is the convolution of the corresponding conditional i ˇ densities p i (·|ME = mE) of m ; i = 1,..., k. Denotem ˜ i ≡ mE,i − I(i = k) andm ˜ S,i ≡ mS S mS,i −I(i = k). Note thatm ˜ i is the number of type-i firms active when continuation decisions are made in the current period, excluding the firm of interest; andm ˜ S,i is the number of firms who have type-i in the current period and continue to next period, excluding the firm of interest. Then, for mS,i such that 0 ≤ mS,i ≤ mE,i, we have that

m˜ i m˜ S,i m˜ i−m˜ S,i p i (·|ME = mE) = aS(mE, c, i) (1 − aS(mE, c, i)) mS m˜ S,i

Computing the expectation in (2) requires the distribution of (N 0,M 0,C0,K0) conditional on MS,C,K, given that all potential entrants use the common strategy aE. Denote the 0 0 0 density of this distribution with p (·|MS,C,K). Conditional on (N ,C ), M is independent 0 0 0 0 0 of (K ,MS,C,K); conditional on (K ,MS,K), N is independent of C ; conditional on C, C is 0 0 independent of (K ,MS,C,K); and conditional on K, K is independent of HS. Consequently,

0 0 0 0 0 0 0 0 0 0 0 0 p (n , m , c , k |MS = mS,C = c, K = k) =pM (m |N = n ,C = c ) × pN (n |K = k ,MS = mS,K = k) 0 × q(c |C = c) × Πk0k.

0 0 0 Here, pM (·|N ,C ) is the density of next period’s post-entry market structure M , conditional 0 0 0 on next period’s pre-entry market structure N and demand state C . And, pN (·|K ,MS,K)

10 0 is the density of next period’s pre-entry market structure N conditional on MS, given that the ﬁrm of interest survives with productivity type K0. First, note that  1 − a (m0 + ι , c0) if M 0 = n0;  E 1  0 0 0 0 0 0 0 0 0 0 0  (1 − aE(m + (m1 − n1 + 1)ι1, c )) if m1 > n1 pM (m |N = n ,C = c ) = 0 0 Qm1−n1 0 0 0 0 0 0  × j=1 aE(n + jι1, c ) and m2 = n2, . . . , mˇ = nˇ;  k k  0 otherwise.

0 0 P i i Next, consider pN (·|K ,MS,K). Decompose n = i n , with n the contribution to next period’s pre-entry market structure by the mS,i firms who are of type-i in the previous period 1 kˇ and choose to continue. Then, because N ,. . . , N are independent conditional on MS and 0 0 given that the firm of interest survives with productivity type K , pN (·|K ,MS,K) is the 0 i ˇ convolution of the corresponding conditional densities pN i (·|K ,MS,K) of N ; i = 1,..., k. i 0 i Denoten ˜ (k, k ) ≡ n − I(i = k)ιk0 . This is the contribution of the mS,i firms excluding the firm of interest, to next period’s pre-entry market structure. Then, for mS such that m˜ S,i ≥ 0, we have that

kˇ Pkˇ i 0 0 n˜ (k, k ) n˜i (k,k0) k 0 0 Y m=i m i0 p i n |K = k ,M = m ,K = k = Π 0 N S S ˜ i 0 i i N 0 (k, k ) i0=i i

˜ i i P i if Nm = 0 for all m < i,n ˜m ≥ 0 for all m ≥ i, and i0 n˜i0 ≤ m˜ i; and zero otherwise.

II.2 Constructing the Type Transition Matrices in Matlab

II.2.1 The Problem

Given any finitem ˇ and kˇ and a kˇ × kˇ transition matrix Π, or the triple (k,ˇ m,ˇ Π), we need to compute all the transition matrices for 1, 2,..., mˇ -firm market structures, conditioning on all realized exits and one surviving firm’s type transition. Since any single firm’s type transition is characterized by Π, the non-trivial part of this problem is computing all the transition matrices for 1, 2,..., mˇ − 1-firm market structures. W.L.O.G., we discuss how to constructm ˇ such matrices for the triple (k,ˇ mˇ + 1, Π). For every ordering of all possible market structures with m firms, m ∈ {1,..., mˇ }, there is a representation of transition matrix corresponding to that ordering. We henceforth focus on the transition matrices for OL-ordered market structures. For any m, we denote the transition matrix as Πm.

11 II.2.2 The Dimensionality

For the triple (k,ˇ mˇ + 1, Π), we know that if there are m surviving ﬁrms, the OL-ordered ˇ sequence of all possible market structures has (m+k−1)! elements. Therefore, Πm’s dimension m!(kˇ−1)! ˇ ˇ is (m+k−1)! × (m+k−1)! . m!(kˇ−1)! m!(kˇ−1)!

II.2.3 Recursive Construction of Πm

We recursively construct Πm using Πm−1 and Π1, for all 2 ≤ i ≤ mˇ . Note that Π1 = Π. To describe the construction, we very often use examples. We use Italic to distinguish the discussion on general case and the discussion on an example. The link between an element in Πm and the elements in Πm−1 and Π1 is explained below.

(i). The (a, b) element in Πm corresponds to a transition probability from an initial m-firm market which has index a in the m-firm OL sequence to a destined m-firm market with index b.

(ii). Suppose that i is the index for the highest type in the initial market a. Taking out one type-i firm from the a leaves an initial m − 1-firm market structure. Suppose that this market structure has index c in the m − 1-firm OL sequence.

(iii). Next, suppose that the type-i ﬁrm transits to one of the possible types j in the destined 1 market b. This transition has probability Πi,j.

(iv). Excluding this type-j firm from the destined market leaves a destined m−1-firm market. Suppose that this market has index d in the m − 1-firm OL sequence.

(v). The transition between the initial and the destined m − 1-ﬁrm markets is characterized m−1 by Πc,d .

(vi). The transition from the initial m-ﬁrm market to the destined one then has the probability

m X m−1 Πa,b = Πi,jΠc,d . j:bj >0

1Note that we also consider the impossible regression in the types here and throughout this notes. So, we consider all j’s including those are lower than i. In such cases, Πi,j = 0.

12 Example Suppose that kˇ = 3, mˇ = 2. In slightly abused notations, we denote the types as L, M, H. Π2 is a 6 × 6 matrix. Now, take its (2, 3) element as an example to demonstrate the above procedure.

(i). This (2, 3) element corresponds to the transition from the market HM to HL.

(ii). Taking out the firm with the highest type H from the initial market HM leaves an initial 1-firm market M, which has index 2 in the 1-firm OL sequence.

(iii). Suppose that the H ﬁrm transits to H in the destined market. This transition has

probability Π1,1.

(iv). Excluding the H firm from the destined market leaves a destined 1-firm market L, which has index 3 in the 1-firm OL sequence.

(v). The transition between the initial and the destined 1-ﬁrm market M and L is charac- 1 terized by Π2,3.

(vi). Note that H can also transit to L (with 0 probability), the transition from the initial market HM to the destined HL then has the probability

2 1 1 Π2,3 = Π1,1Π2,3 + Π1,3Π2,1.

In short-hand notations, we rewrite the equation Πm = P Π Πm−1 using indices ab j:bj >0 ij cd only: (a, b) := P (i, j) × (c, d), with the understanding that (a, b) always indexes the j:bj >0 element in Πm,(i, j) in Π, and (c, d) in Πm−1. We connect these indices to the objects that they index.

(i). i indexes the highest type in the market a. Therefore, for each given a, i is unique. This implies that in each row of Πm, all entries share the same i.

(ii). All j’s indicate all possible types in market b. Therefore, in each column of Πm, all entries share the same j’s. For any (a, b) entry, there are at most kˇ possible values of j.

(iii). c indexes the m − 1 market structure resulted by subtracting a type-i ﬁrm from the market a. Therefore, for each (a, b), c is unique and in each row of Πm, all entries share the same c.

13 (iv). All d’s index all possible m − 1 market structures resulted by subtracting a type-j ﬁrm from the market b. Therefore, in each column of Πm, all entries share the same d’s. For any (a, b) entry, there are at most kˇ possible values of d.

Henceforth, we call i the ﬁrst index, all j’s the second indices, c the third, and d’s the fourth. One may have already developed some intuition that there are regularity patterns in these indices, which can be used to vectorize the calculation of Πm. Next, we make the regularity pattern visible to intellectual eyes by an example.

Example As an example, we write Π2 for kˇ = 3, mˇ = 2 using the indices representation. Again, bear in mind that the ﬁrst two indices index the element in Π while the last two index the element in Π1.

14 HH HM HL MM ML LL HH (1, 1) × (1, 1) (1, 1) × (1, 2) (1, 1) × (1, 3) +(1, 2) × (1, 1) (1, 2) × (1, 2) (1, 2) × (1, 3) +(1, 3) × (1, 1) +(1, 3) × (1, 2) (1, 3) × (1, 3) HM (1, 1) × (2, 1) (1, 1) × (2, 2) (1, 1) × (2, 3) +(1, 2) × (2, 1) (1, 2) × (2, 2) (1, 2) × (2, 3) +(1, 3) × (2, 1) +(1, 3) × (2, 2) (1, 3) × (2, 3) HL (1, 1) × (3, 1) (1, 1) × (3, 2) (1, 1) × (3, 3) +(1, 2) × (3, 1) (1, 2) × (3, 2) (1, 2) × (3, 3) +(1, 3) × (3, 1) +(1, 3) × (3, 2) (1, 3) × (3, 3) MM (2, 1) × (2, 1) (2, 1) × (2, 2) (2, 1) × (2, 3) +(2, 2) × (2, 1) (2, 2) × (2, 2) (2, 2) × (2, 3) +(2, 3) × (2, 1) +(2, 3) × (2, 2) (2, 3) × (2, 3) 15 ML (2, 1) × (3, 1) (2, 1) × (3, 2) (2, 1) × (3, 3) +(2, 2) × (3, 1) (2, 2) × (3, 2) (2, 2) × (3, 3) +(2, 3) × (3, 1) +(2, 3) × (3, 2) (2, 3) × (3, 3) LL (3, 1) × (3, 1) (3, 1) × (3, 2) (3, 1) × (3, 3) +(3, 2) × (3, 1) (3, 2) × (3, 2) (3, 2) × (3, 3) +(3, 3) × (3, 1) +(3, 3) × (3, 2) (3, 3) × (3, 3) To illustrate the regularity in the above matrix, the first trick is to introduce an auxiliary ˇ ”impossible destined market structure”, which possesses index (m+k−1)! +1 in the OL sequence m!(kˇ−1)! of m-firm market structures. Its impossibility means that no m-firm market structure can transit to it. For instance, when m = 1, this market structure has index 4. To accommodate such impossible destined market structure, we can expand Π1 by a fourth column of zeros, so 1 Πi,4 = 0, i = 1, 2, 3. Then, we can rewrite the above matrix as

16 HH HM HL MM ML LL HH (1, 1) × (1, 1) (1, 1) × (1, 2) (1, 1) × (1, 3) (1, 1) × (1, 4) (1, 1) × (1, 4) (1, 1) × (1, 4) +(1, 2) × (1, 4) +(1, 2) × (1, 1) +(1, 2) × (1, 4) +(1, 2) × (1, 2) +(1, 2) × (1, 3) +(1, 2) × (1, 4) +(1, 3) × (1, 4) +(1, 3) × (1, 4) +(1, 3) × (1, 1) +(1, 3) × (1, 4) +(1, 3) × (1, 2) +(1, 3) × (1, 3) HM (1, 1) × (2, 1) (1, 1) × (2, 2) (1, 1) × (2, 3) (1, 1) × (2, 4) (1, 1) × (2, 4) (1, 1) × (2, 4) +(1, 2) × (2, 4) +(1, 2) × (2, 1) +(1, 2) × (2, 4) +(1, 2) × (2, 2) +(1, 2) × (2, 3) +(1, 2) × (2, 4) +(1, 3) × (2, 4) +(1, 3) × (2, 4) +(1, 3) × (2, 1) +(1, 3) × (2, 4) +(1, 3) × (2, 2) +(1, 3) × (2, 3) HL (1, 1) × (3, 1) (1, 1) × (3, 2) (1, 1) × (3, 3) (1, 1) × (3, 4) (1, 1) × (3, 4) (1, 1) × (3, 4) +(1, 2) × (3, 4) +(1, 2) × (3, 1) +(1, 2) × (3, 4) +(1, 2) × (3, 2) +(1, 2) × (3, 3) +(1, 2) × (3, 4) +(1, 3) × (3, 4) +(1, 3) × (3, 4) +(1, 3) × (3, 1) +(1, 3) × (3, 4) +(1, 3) × (3, 2) +(1, 3) × (3, 3) MM (2, 1) × (2, 1) (2, 1) × (2, 2) (2, 1) × (2, 3) (2, 1) × (2, 4) (2, 1) × (2, 4) (2, 1) × (2, 4) +(2, 2) × (2, 4) +(2, 2) × (2, 1) +(2, 2) × (2, 4) +(2, 2) × (2, 2) +(2, 2) × (2, 3) +(2, 2) × (2, 4) +(2, 3) × (3, 4) +(2, 3) × (3, 4) +(2, 3) × (2, 1) +(2, 3) × (3, 4) +(2, 3) × (2, 2) +(2, 3) × (2, 3) 17 ML (2, 1) × (3, 1) (2, 1) × (3, 2) (2, 1) × (3, 3) (2, 1) × (3, 4) (2, 1) × (3, 4) (2, 1) × (3, 4) +(2, 2) × (3, 4) +(2, 2) × (3, 1) +(2, 2) × (3, 4) +(2, 2) × (3, 2) +(2, 2) × (3, 3) +(2, 2) × (3, 4) +(2, 3) × (3, 4) +(2, 3) × (3, 4) +(2, 3) × (3, 1) +(2, 3) × (3, 4) +(2, 3) × (3, 2) +(2, 3) × (3, 3) LL (3, 1) × (3, 1) (3, 1) × (3, 2) (3, 1) × (3, 3) (3, 1) × (3, 4) (3, 1) × (3, 4) (3, 1) × (3, 4) +(3, 2) × (3, 4) +(3, 2) × (3, 1) +(3, 2) × (3, 4) +(3, 2) × (3, 2) +(3, 2) × (3, 3) +(3, 2) × (3, 4) +(3, 3) × (3, 4) +(3, 3) × (3, 4) +(3, 3) × (3, 1) +(3, 3) × (3, 4) +(3, 3) × (3, 2) +(3, 3) × (3, 3) The second trick towards detecting the regularity is to partition the matrix into kˇ row- blocks. In each row-block, all rows correspond to initial market structures that share the same highest type. In the above matrix, row 1-3 correspond to initial market structures whose highest type is H, row 4-5 M, and row 6 L. Because of OL, the index of the row-block is also the index for the highest type. So, in the above matrix, the first block connects to type 1, the second block to type 2, and the third block to type 3. The first row-block of any Πm matrix contains all initial market structures that has one ˇ type-kˇ firm and m−1 other firms with any type. There are (m−1+k−1)! such market structures. (m−1)!(kˇ−1)! ˇ So the length of the first block is just (m−1+k−1)! . The second block contains all initial market (m−1)!(kˇ−1)! structures that has one type-kˇ − 1 firm and m − 1 other firms with type no better than kˇ − 1. ˇ There are (m−1+k−2)! such market structures. So is the length of the second block. In total, (m−1)!(kˇ−2)! ˇ there are kˇ such blocks. The t-th block has the length (m−1+k−t)! . (m−1)!(kˇ−t)! Now each entry of the matrix has four columns of indices. All columns have the same length kˇ (3 in this example). Next, we present the regularity on these columns.

(i). Recall that the ﬁrst index represents the highest type in the initial market structure. Also recall that in each row-block, all initial market structures share the same highest type. Therefore, the column of ﬁrst indices in each entry has a single value, which is simply the index of the row-block that this entry is in. Therefore, it remains unchanged for every row in a same block.

(ii). Recall that the second indices represent all the possible types in the destined market structure. After the introduction of the impossible market structure, the column of the second indices in each entry is simply (1,..., kˇ).

(iii). Recall that the third index represents the m−1 market structure resulted by subtracting a highest type firm from the initial market structure. Therefore, the column of the third indices has a single value and remains unchanged for every entry in a same row. In each row, this value equals the index of the m − 1 market structure resulted by subtracting a highest type from the initial market structure. In the current example, in row 1, the 1-firm market structure resulted by subtracting H from HH is H, which has index number 1 in 1-firm OL sequence. So, in the first row, the third index is 1. In row 2, the 1-firm market structure resulted by subtracting H from HM is M. So, in the first row, the third index is 2. Observing the following facts

(a) Within each block, this index increases by 1 each row.

18 (b) The last row in each block corresponds to the most inferior market structure in the OL sense. Hence, this index in the last row of each block must equal to the length of the OL sequence of the m − 1-ﬁrm market structures. In the current example, the length is 3, which is the value of the third index in row 3, 5, 6. ˇ (c) The t-th block has the length (m−1+k−t)! . (m−1)!(kˇ−t)!

ˇ We can conclude that in the t-th row-block, the third index grows from (m−1+k−1)! − (m−1)!(kˇ−1)! ˇ ˇ (m−1+k−t)! + 1 to (m−1+k−1)! row by row. (m−1)!(kˇ−t)! (m−1)!(kˇ−1)! (iv). Recall that the fourth indices represents the m − 1 market structure resulted by subtracting a highest type ﬁrm from the destined market structure. Therefore, the column of the fourth indices remains unchanged for every entry in a same column. The regularity pattern of this column is more subtle than any of the above columns. We further explore it. We write down this column in the above example HH HM HL MM ML LL 1 2 3 4 4 4 4 1 4 2 3 4 4 4 1 4 2 3 This matrix of the fourth indices can be engineered from the following 0-1 matrix. HH HM HL MM ML LL 1 1 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 We can transform the 1’s in each row of the above matrix into the ordinals of 1’s (the ﬁrst 1 stays 1, the second 1 is transformed to 2, the third to 3) and the 0’s into 4 to go back to the matrix of the fourth indices. This transformation is unique and can always be done for any matrix of the fourth indices. Hence we focus on constructing the later matrix, which we simply call the indexing matrix. Since the fourth indices are related to the destined market structures, we construct the indexing matrix by exploring Πm from its column dimension. Now, we introduce the third trick. We partition the Πm matrix into kˇ column-blocks. Analogously to the row- blocks, in each column-block, all columns correspond to destined market structures that share the same highest type. In the example matrix, column 1-3 correspond to destined market structures whose highest type is H, column 4-5 M, and column 6 L. Again, the index of the column-block is also the index for the highest type. So, in the

19 above matrix, the ﬁrst block connects to type 1, the second block to type 2, and the third block to type 3. Observing the following facts

(a) The indexing matrix for m = 1 is a kˇ × kˇ identity matrix. (b) The indexing matrix has kˇ rows. Its (e, f) element indicates if the destined market structure f has a type-e firm. If it does, then the (e, f) element of the indexing matrix is 1. Otherwise it is 0. In the above example, the (1, 1) element of the indexing matrix is 1, because the market structure HH contains a type-H firm. The (2, 1) element is 0, because the market structure HH does not contain a type- M firm. (c) The indexing matrix can also be partitioned into kˇ column-blocks. (d) In its t-th column-block, since the highest type in the destined market structure is t, the first t − 1 rows of the indexing matrix in this block are all 0’s and the t-th row is full of 1’s. In the above example, the first row is full of 1’s in block 1 and full of 0’s in block 2. (e) In its t-th column-block, since the m − 1-firm market structures resulted by subtracting the highest type firm from the destined market structure are the ˇ ˇ ˇ (m−1+k−1)! − (m−1+k−t)! + 1 to (m−1+k−1)! destined market structures in Πm−1, (m−1)!(kˇ−1)! (m−1)!(kˇ−t)! (m−1)!(kˇ−1)! ˇ from the t + 1-th row onward, the indexing matrix is identical to the (m−1+k−1)! − (m−1)!(kˇ−1)! ˇ ˇ (m−1+k−t)! + 1 to (m−1+k−1)! columns of the indexing matrix corresponding to (m−1)!(kˇ−t)! (m−1)!(kˇ−1)! m − 1. In column-block 1 in the above example, the 1-firm market structures are H,M, and L, which are the 1,2, and 3 destined market structures of Π1. Hence, from the second row onwards in block 1, the indexing matrix is identical to the 1,2, and 3 columns of the indexing matrix for m = 1, which is a 3 × 3 identity matrix. In column-block 2 in the above example, the 1-firm market structures are M and L, which are the 2, and 3 destined market structures of Π1. Hence, the third row in block 2 of the indexing matrix is identical to the third row and the 2 and 3 columns of the indexing matrix for m = 1.

With all the regularity patterns pointed out as above, we create a Matlab function to generate all the transition matrices.

20 II.2.4 The typetransition.m Function

The matlab function typetransition.m takes the triple (k,ˇ m,ˇ Π) as input, and produces a ˇ ˇ (m ˇ +k−1)! × ( (m ˇ +k−1)! + 1) × mˇ array, in which each page contains a transition matrix and the mˇ !(kˇ−1)! mˇ !(kˇ−1)! ˇ page number m indicates the number of firms. On each page, the first (m+k−1)! rows and the m!(kˇ−1)! ˇ first (m+k−1)! columns form the transition matrix for the m-firm market. m!(kˇ−1)! This function has several layers of loops. The most outside loop runs from m = 2 to m =m ˇ . Within this loop, for each given m, the indexing matrix is first created and than transformed to the matrix of the fourth indices. Then, we use the above mentioned regularity patterns to construct the other three columns of indices and compute the transition matrix Πm row-by-row. Last, a few words on the computational speed. When kˇ =m ˇ = 7, the transition matrix is computed within 3 seconds. When kˇ =m ˇ = 8, around 60 seconds. When kˇ =m ˇ = 9, a normal PC runs out of memory.

II.3 Computing All Renegotiation-proof Natural Markov-Perfect Equilibria

In this appendix, we first show that when C is discrete, we can compute all renegotiation- proof natural Markov-perfect equilibria. Then, we discuss how to modify Algorithm2 to compute all such equilibria. We have seen in Section ?? that the multiplicity of renegotiation-proof equilibria comes from the multiple mixing probabilities that can solve (11). Therefore, to compute any single equilibrium using Algorithm2, we always need to select the probability corresponding to this equilibrium. To this end, we introduce a flexible selection mechanism which enables us to do so. kˇ A selection rule of such mechanism is summarized by Γ : Z? × [ˆc, cˇ] × K → {1,..., mˇ }. It works as follows. Suppose that a renegotiation-proof Markov-perfect equilibrium exists, and aS(m, c, k), as a mixing probability, can take σ(m, c, k) values. Sort all these possible values in a (weakly) descending sequence. Then, we use Γ to uniquely pin down aS(m, c, k) by setting aS(m, c, k) = min{σ(m, c, k), Γ(m, c, k)}-th possible value in this sequence. To give an example of Γ, if for any (m, c, k), Γ(m, c, k) = 1. Then, we always pick the first one in the sequence or the largest probability as the survival rule. With a pre-specified Γ, we can modify Procedure3 to include this mechanism and compute a renegotiation-proof Markov-perfect equilibrium.

21 mi, ki

i Specify c ∈ [ˆc, cˇ] and m ∈ i , ∀0 ≤ j ≤ mˇ − |m | mm +jιki

i Yes αS(m, c, k ) = m i = 1 ? k I[w(m, c, ki) > 0]

i Yes wE(m, c, k ) α (m, c, ki) = 1 > 0? S

No Find all p’s ∈ [0, 1) satisfying, m −1 ki m i − 1 X m i −1−j j k i (1−p) k p w (m−(m i −1−j)ι i , c, k ) = 0. j S k k j=0

i i No wS(m , c, k ) Yes > 0?

Sort p’s and 0 in Sort p’s in a de- a decreasing array creasing array

i σ(m, c, k ) ← Length of the array, αS(m, c, k(m)) ← the min{σ(m, c, ki), Γ(m, c, ki)}-th element of this array

α (m, c) = I[w(m + E Yes i jι1, c, 1) − ϕ > k = 1 ? 0, ∀0 ≤ j ≤ mˇ − |m|]

CONTINUE

Procedure 3: Calculation of Candidate Entry/Survival22 Rule for the General Model, Non- Monotone Payoffs Because the number of possible mixing probabilities is bounded by the number of roots of the polynomial in equation (11), which is in turn bounded by the polynomial’s order. In the general model, the highest order of any polynomial in equation (11) ism ˇ . Thus, from the definition of Γ, it is clear that if C is a discrete variable, the number of distinct Γ mappings is finite. Therefore, we can compute all renegotiation-proof natural Markov-perfect equilibria for the general model by implementing Algorithm2 repeatedly for all possible Γ’s. Although this procedure can be completely parallelized, it is still computationally cumbersome for large m,ˇ kˇ and large number of possible realizations of C. Practically, we can reduce the computational burden by avoiding running the algorithm for ”redundant” Γ’s. For some (m, c, k) ∈ S such that σ(m, c, k) < mˇ , suppose that under a selection rule Γ, Γ(m, c, k) = σ(m, c, k). Then any Γ˜ with Γ(˜ m, c, k) > σ(m, c, k) and Γ(n, d, g) = Γ(˜ n, d, g), for all (n, d, g) 6= (m, c, k) selects the same Markov-perfect equilibrium as Γ. Therefore, all such Γ˜ (there arem ˇ − σ(m, c, k) of them) are redundant, provided that we have run the algorithm for Γ. This suggests that to find all the renegotiation-proof natural equilibria in a computationally efficient way, we should run the algorithm with no pre-specified Γ but ”branch” the algorithm once multiplicity arises. To be more specific, after starting the algorithm, once we reach a (m, c, k) such that σ(m, c, k) > 1, we create

σ(m, c, k) branches with αS(m, c, k) set diﬀerently. Diﬀerent branches then can be computed in parallel. The same branching exercise is done for each parallel session when a new state with multiple choices emerges.

III Computational Details

III.1 Pencil-and-Paper Computation behind Figure3

One can compute the example graphed in Figure3 using numerical method, such as value function iteration. Alternatively, this example can be computed exactly using only pencil and paper. This appendix contains the details. Begin with characterizing a duopolist’s payoﬀ function which satisﬁes ( 0 if Cc ≤ c2, vE(2, c) = c (1−λ)( 2 π(2)−κ)+λve(2) β 1−β(1−λ) if c > c2,

23 where  cˆ if v (2, cˆ) > 0,  E c2 ≡ cˇ if vE(2, cˇ) < 0,   max{c|vE(2, c) = 0} otherwise, and 1 cˆ +c ˇ Z cˇ v (2, c) v(2) = π(2) − κ + E dC. e 2 2 (ˇc − cˆ) c2 We want to calculate the continuation and entry thresholds. Ifcπ ˆ (2)/2 > κ, firms always earn positive payoff no matter which C is drawn, then vE(2, cˆ) > 0 and we settle with the corner solution c2 =c ˆ. Ifcπ ˆ (2)/2 < κ, we can normalize [ˆc, cˇ] to [0, 1] with the transformations C − cˆ C0 ≡ , cˇ − cˆ π0(2) ≡ π(2) (ˇc − cˆ) , cˆ κ0 ≡ κ − π(2). 2 We then proceed by first considering two corner solutions.

If vE(2, 0) > 0, no duopolist will ever exit the market and they expect to earn average proﬁt from perpetual operation if demand state switches. So ve(2) in this case is 1 π0(2) v(2) = − κ0 . e 1 − β 4

Then vE(2, 0) > 0 is

β(1 − λ) π0(2) βλ π0(2) v (2, 0) = 0 × − κ0 + − κ0 > 0 E 1 − β(1 − λ) 4 (1 − β(1 − λ)) (1 − β) 4

Simplify the above expression we produce the necessary and suﬃcient condition for the corner solution c2 = 0 as

γ − γβ + γλβ − λ < 0,

4κ0 with γ ≡ π0(2) . If vE(2, 1) < 0, it is not possible for both duopolist to always choose continuation, so

β(1 − λ) π0(2) βλ π0(2) v (2, 1) = − κ0 + − κ0 < 0. E 1 − β(1 − λ) 2 1 − β(1 − λ) 4

24 Simplify this expression we obtain the necessary and suﬃcient condition for the corner solution c2 = 1 as γ > 2 − λ.

Now we proceed to calculate the interior solution. Substituting the lower branch of vE(2, c) into the expression for ve(2) produces π0(2) Z 1 v(2) = − κ + vE(2,C)dC e 4 0 c2 π0(2) β(1 − λ) π0(2) βλv(2) = − κ0 + (1 − (c0 )2) − κ0(1 − c0 ) + e (1 − c(6)0 ). 4 1 − β(1 − λ) 4 2 2 1 − β(1 − λ) 2 0 In addition c2 by deﬁnition must satisfy c0 (1 − λ) 2 π0(2) − κ + λv(2) = 0, 2 e so c0 1 − λ v(2) = κ0 − 2 π0(2) . (7) e 2 λ Combine (6) and (7) and then rearrange to obtain β(1 − λ) π0(2) π0(2) 1 − λ β(1 − λ) π0(2) (c0 )2 + − c0 1 − β(1 − λ) 4 2 2 λ 1 − β(1 − λ) 2 2 (8) π0(2) β(1 − λ) π0(2) 1 − λ + − κ0 + − κ0 = 0. 4 1 − β(1 − λ) 4 λ

0 2 (8) is a quadratic equation in c2 of the form ax + bx + c = 0 with β(1 − λ) π0(2) a = , 1 − β(1 − λ) 2 π0(2) 1 − λ β(1 − λ) π0(2) b = − , 2 λ 1 − β(1 − λ) 4 π0(2) β(1 − λ) π0(2) 1 − λ c = − κ0 + − κ0. 4 1 − β(1 − λ) 4 λ √ −b± b2−4ac Use x1,2 = 2a to calculate the roots for (8) as ! 1 r(1 − λ − β + γλβ)(1 − β(1 − λ)) c01 = β − 1 + , and (9) 2 λβ 1 − λ ! 1 r(1 − λ − β + γλβ)(1 − β(1 − λ)) c02 = β − 1 − (10) 2 λβ 1 − λ

25 02 Because β < 1, c2 ∈/ [0, 1]. The analysis above tell us that the necessary condition for λ 01 an interior solution to exit is 2 − λ > γ > 1−β(1−λ) . We use this to check that c2 is in the admissible set (0, 1). 01 λ Apparently c2 is increasing in γ. Substitute γ = 1−β(1−λ) into (9) to calculate the lower 01 bound for c2 as 1 c01 = β − 1 + p(1 − β)2 = 0. 2 λβ

01 Substitute γ = 2 − λ into (9) to calculate the upper bound for c2 as 1 c01 = β − 1 + p(1 − β(1 − λ))2 = 1. 2 λβ

λ 0 Therefore, when 2 − λ > γ > 1−β(1−λ) , the interior solution for c2 is given by (9). Finally, 0 we restore c2 by using c2 = c2(ˇc − cˆ) +c ˆ.

Before we turn to the analysis for a monopolist, we calculate c2 which is deﬁned as  cˆ if v (2, cˆ) > ϕ,  E c2 ≡ cˇ if vE(2, cˇ) < ϕ,   max{c|vE(2, c) − ϕ = 0} otherwise,

After obtaining c2, using (7), we can calculate ve(2). Then by setting vE(2, c2) = ϕ, we calculate c2 as 2 ϕ(1 − β(1 − λ)) − λβv(2) c = e + κ . (11) 2 π(2) β(1 − λ) With this in hand, we can deﬁne a monopolist’s payoﬀ ( 0 if c ≤ c1, vE(1, c) = (1−λ)(cπ(1)−κ)+λve(1) β 1−β(1−λ) if c > c1, where  cˆ if v (1, cˆ) > 0,  E c1 ≡ cˇ if vE(1, cˇ) < 0,   max{c|vE(1, c) = 0} otherwise, and

cˆ +c ˇ Z cˇ v (2,C) Z c2 v (1,C) v(1) = π(1) − κ + E dC + E dC. e 2 (ˇc − cˆ) (ˇc − cˆ) c2 c1

26 Ifcπ ˆ (1) > κ, ﬁrm always earns positive payoﬀ no matter which C is drawn, then vE(1, cˆ) >

0 and we settle with the corner solution c1 =c ˆ. Ifcπ ˆ (1) < κ, we can normalize [ˆc, cˇ] to [0, 1] with the transformations C − cˆ C0 ≡ , cˇ − cˆ π0(1) ≡ π(1) (ˇc − cˆ) , κ0 ≡ κ − cπˆ (1).

Because Z 1 0 0 vE(2,C )dC 0 c2 is readily computable by using the deﬁnition of vE(2,C). We hereafter denote its value by ν. Similarly to the duopolist case, we then proceed by ﬁrst considering two corner solutions. If vE(1, 0) > 0, no monopolist will exit the market and we can calculate ve(1) in this case as 1 (c0 )2 β(1 − λ) β(1 − λ)c0 βλc0 v(1) = π0(1) − κ0 + ν + 2 π0(1) − 2 κ0 + 2 v(1). e 2 2 1 − β(1 − λ) 1 − β(1 − λ) 1 − β(1 − λ)e This gives

0 0 0 2 0 0 0 (π (1)/2 − κ + ν) (1 − β(1 − λ)) + β(1 − λ) ((c2) π (1)/2 − c2κ ) ve(1) = 0 . 1 − β + λβ(1 − c2)

Then vE(1, 0) > 0 is β(1 − λ) βλ v (1, 0) = (−κ0) + E 1 − β(1 − λ) (1 − β(1 − λ)) 0 0 0 2 0 0 0 (π (1)/2 − κ + ν) (1 − β(1 − λ)) + β(1 − λ) ((c2) π (1)/2 − c2κ ) × 0 . 1 − β + λβ(1 − c2) Simplify the above expression we produce the necessary and suﬃcient condition for the corner solution c1 = 0 as (c0 )2(1 − λ)β γ0 < 2 + 1, (1 − β(1 − λ))

0 2 κ0 with γ ≡ π0(1) λ − ν . If vE(1, 1) < 0, a monopolist will never remain in the market and entry will never happen, so β(1 − λ) βλ π0(1) v (1, 1) = (π0(1) − κ0) + − κ0 < 0. E 1 − β(1 − λ) 1 − β(1 − λ) 2

27 Simplify this expression we obtain the necessary and suﬃcient condition for the corner solution c1 = 1 as 2 − λ π0(1) < κ0. 2 Because ν = 0 in this case, we can add λν to the the right hand side without changing the result. Divide both sides by π0(1)λ/2 2 − λ γ0 > . λ Now we proceed to calculate the interior solution. Substituting the lower branch of vE(1, c) into the expression for ve(1) produces 0 0 Z c2 π (1) 0 v(1) = − κ + ν + vE(1,C)dC e 2 0 c1 π0(1) β(1 − λ) π0(1) = − κ0 + ν + ((c0 )2 − (c0 )2) − κ0(c0 − c0 ) 2 1 − β(1 − λ) 2 2 1 2 1 βλv(1)(c0 − c0 ) + e 2 1 . (12) 1 − β(1 − λ) 0 In addition c1 by deﬁnition must satisfy

0 0 0 (1 − λ)(c1π (1) − κ ) + λve(1) = 0, so 1 − λ v(1) = (κ0 − c0 π0(1)) . (13) e 1 λ Combine (12) and (13) and then rearrange to obtain β(1 − λ) π0(1) 1 − λ β(1 − λ)c0 (c0 )2 + π0(1) − 2 π0(1) c0 1 − β(1 − λ) 2 1 λ 1 − β(1 − λ) 1 (14) π0(1) β(1 − λ)(c0 )2 π0(1) 1 − λ + − κ0 + ν + 2 − κ0 = 0. 2 1 − β(1 − λ) 2 λ √ 0 2 −b± b2−4ac (14) is a quadratic equation in c1 of the form ax + bx + c = 0. Use x1,2 = 2a to calculate the roots for (14) as s β − 1 (ξ2 − 2ηξc0 − η) + ηγ0 c01 = (c0 − 1) + + 2 , and (15) 1 2 λβ η2 s β − 1 (ξ2 − 2ηξc0 − η) + ηγ0 c02 = (c0 − 1) + + 2 (16) 1 2 λβ η2

28 with 1−λ ξ = λ , β(1−λ) η = 1−β(1−λ) . 0 02 Because β < 1 and c2 < 1, c1 ∈/ [0, 1]. The analysis above tell us that the necessary 0 2 2−λ 0 (c2) (1−λ) condition for an interior solution to exit is λ > γ > (1−β(1−λ))λ + 1. We use this to check 01 that c1 is in the admissible set (0, 1). 0 2 01 0 0 (c2) (1−λ) Apparently c1 is increasing in γ . Substitute γ = (1−β(1−λ))λ + 1 to calculate the lower 01 bound for c1 . s β − 1 ξ2 − 2ηξc0 − η + η + η2(c0 )2 c01 = c0 − 1 + + 2 2 = 0. 1 2 λβ η2 which can be simpliﬁed to s ξ (ξ − ηc0 )2 c01 = c0 − + 2 = 0. 1 2 η η2

2−λ 01 Substitute γ = λ into (15) to calculate the upper bound for c1 . s 1 (1 − β(1 − λ) − 2λβc0 + 2λβ)(1 − β(1 − λ)) c01 = (c0 − 1) + (β − 1) + 2 . 1 2 λβ λ2β2 Remember when monopolist exit the market for sure, no duopolist will choose to survive. 0 So c2 = 1 and above equation simpliﬁes to s 1 (1 − β(1 − λ))2 c01 = (β − 1) + = 1. 1 λβ λ2β2

0 2 2−λ 0 (c2) (1−λ) 0 Therefore, when λ > γ > (1−β(1−λ))λ + 1, the interior solution for c1 is given by (15). 0 Finally, we restore c1 by using c1 = c1(ˇc − cˆ) +c ˆ.

Last thing is to calculate c1, which is deﬁned as  cˆ if v (1, cˆ) > ϕ,  E c1 ≡ cˇ if vE(1, cˇ) < ϕ,   max{c|vE(1, c) − ϕ = 0} otherwise,

After obtaining c1, using (13), we can calculate ve(1). Then by setting vE(1, c1) = ϕ, we calculate c1 as 1 ϕ(1 − β(1 − λ)) − λβv(1) c = e + κ . (17) 1 π(1) β(1 − λ)

Because firms’ payoffs are linear functions in C, knowing c1, c1, c2, c2 is sufficient for determining the payoffs for duopolist and monopolist. The calculation is completed.

29 III.2 Computing the Example of Multiple Equilibria in Section 4.2

Note that this model is deterministic. We compute some important equilibrium payoﬀs which account for equilibrium multiplicity as below

(i). Since c3πH(3ιH) > κ, if three type-H firms are active in the second period, they can always recover fixed cost and make positive profit by remaining active from the third period onwards. Moreover, if less type-H firms are active in the second period, they

receive higher proﬁt from the third period onward. Therefore, for t ≥ 2, aS(3ιH, ct, H) is a dominant strategy and β(c π (3ι ) − κ) v (3ι , c , H) = 3 H H = 1. E H t 1 − β

(ii). Since c3πL(2ιH + ιL) > κ, c3πH(ιH + 2ιL) > κ and c3πL(ιH + 2ιL) > κ, for the same reason, for t ≥ 2,

vE(2ιH + ιL, ct, L) = vS(2ιH + ιL, ct, L) β(1 − ΠLH) = (c3πL(2ιH + ιL) − κ) 1 − β(1 − ΠLH) β β(1 − ΠLH) + − (c3πH(3ιH) − κ) 1 − β 1 − β(1 − ΠLH) = 0.8167

vE(2ιH + ιL, ct, H) = vS(2ιH + ιL, ct, H)

β(1 − ΠLH) = (c3πH(2ιH + ιL) − κ) 1 − β(1 − ΠLH) β β(1 − ΠLH) + − (c3πH(3ιH) − κ) 1 − β 1 − β(1 − ΠLH) = 1.4

vE(ιH + 2ιL, ct, L) = vS(ιH + 2ιL, ct, L)

1 2 = 2 ((1 − ΠLH) (c3πL(ιH + 2ιL)) 1 − β(1 − ΠLH) +(1 − ΠLH)ΠLH(c3πH(2ιH + ιL) + vE(2ιH + ιL, ct+1, H))

+(1 − ΠLH)ΠLH(c3πL(2ιH + ιL) + vE(2ιH + ιL, ct+1, L)) 2 +ΠLH(c3πH(3ιH) + vE(3ιH, ct+1, L)) − κ) = 1.2881

(iii). Since vE(2ιH + ιL, c2, L) < ϕ and vE(ιH + 2ιL, c2, L) > ϕ, we have that aE(2ιH +

ιL, c2, L) = 0 and aE(ιH + 2ιL, c2, L) = 1. This means that in the second period, two

30 type-L ﬁrms will enter a market occupied by a type-H monopoly, while no ﬁrm will enter a market occupied by two type-H duopoly. Since demand stays constant from the third period on, the market structures at the end of the second period will never be changed. Therefore, for t ≥ 2,

vE(2ιH, ct, H) = vS(2ιH, ct, H) β(c π (2ι ) − κ) = 3 H H 1 − β = 496

vE(ιH + 2ιL, ct, H) = vS(ιH + 2ιL, ct, H)

1 2 = 2 ((1 − ΠLH) (c3πH(ιH + 2ιL)) 1 − β(1 − ΠLH) +2(1 − ΠLH)ΠLH(c3πH(2ιH + ιL) + vE(2ιH + ιL, ct+1, H)) 2 +ΠLH(c3πH(3ιH) + vE(3ιH, ct+1, L)) − κ) = 1.6357

(iv). For a type-H monopolist who is active in the ﬁrst period, the payoﬀ to continuation is

vS(ιH, c1, H) = β ((c2πH(ιH) − κ) + vE(ιH + 2ιL, c2, H)) = −1.1821.

For a type-H duopolist who is active in the ﬁrst period together with another type-H rival, the payoﬀ to continuation, given the rival also continues, is

vS(2ιH, c1, H) = β ((c2πH(2ιH) − κ) + vE(2ιH, c2, H)) = 246.

For a type-H triopolist who is active in the ﬁrst period together with another two type-H rivals, the payoﬀ to continuation, given the rivals also continue, is

vS(3ιH, c1, H) = β ((c2πH(3ιH) − κ) + vE(3ιH, c2, H)) = −1.5.

31 Working Paper Series

A series of research studies on regional economic issues relating to the Seventh Federal Reserve District, and on financial and economic topics.

Risk Taking and the Quality of Informal Insurance: Gambling and Remittances in Thailand WP-07-01 Douglas L. Miller and Anna L. Paulson

Fast Micro and Slow Macro: Can Aggregation Explain the Persistence of Inflation? WP-07-02 Filippo Altissimo, Benoît Mojon, and Paolo Zaffaroni

Assessing a Decade of Interstate Bank Branching WP-07-03 Christian Johnson and Tara Rice

Debit Card and Cash Usage: A Cross-Country Analysis WP-07-04 Gene Amromin and Sujit Chakravorti

The Age of Reason: Financial Decisions Over the Lifecycle WP-07-05 Sumit Agarwal, John C. Driscoll, Xavier Gabaix, and David Laibson

Information Acquisition in Financial Markets: a Correction WP-07-06 Gadi Barlevy and Pietro Veronesi

Monetary Policy, Output Composition and the Great Moderation WP-07-07 Benoît Mojon

Estate Taxation, Entrepreneurship, and Wealth WP-07-08 Marco Cagetti and Mariacristina De Nardi

Conflict of Interest and Certification in the U.S. IPO Market WP-07-09 Luca Benzoni and Carola Schenone

The Reaction of Consumer Spending and Debt to Tax Rebates – Evidence from Consumer Credit Data WP-07-10 Sumit Agarwal, Chunlin Liu, and Nicholas S. Souleles

Portfolio Choice over the Life-Cycle when the Stock and Labor Markets are Cointegrated WP-07-11 Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein

Nonparametric Analysis of Intergenerational Income Mobility WP-07-12 with Application to the United States Debopam Bhattacharya and Bhashkar Mazumder

How the Credit Channel Works: Differentiating the Bank Lending Channel WP-07-13 and the Balance Sheet Channel Lamont K. Black and Richard J. Rosen

Labor Market Transitions and Self-Employment WP-07-14 Ellen R. Rissman

First-Time Home Buyers and Residential Investment Volatility WP-07-15 Jonas D.M. Fisher and Martin Gervais

1 Working Paper Series (continued)

Establishments Dynamics and Matching Frictions in Classical Competitive Equilibrium WP-07-16 Marcelo Veracierto

Technology’s Edge: The Educational Benefits of Computer-Aided Instruction WP-07-17 Lisa Barrow, Lisa Markman, and Cecilia Elena Rouse

The Widow’s Offering: Inheritance, Family Structure, and the Charitable Gifts of Women WP-07-18 Leslie McGranahan

Incomplete Information and the Timing to Adjust Labor: Evidence from the Lead-Lag Relationship between Temporary Help Employment and Permanent Employment WP-07-19 Sainan Jin, Yukako Ono, and Qinghua Zhang

A Conversation with 590 Nascent Entrepreneurs WP-07-20 Jeffrey R. Campbell and Mariacristina De Nardi

Cyclical Dumping and US Antidumping Protection: 1980-2001 WP-07-21 Meredith A. Crowley

Health Capital and the Prenatal Environment: The Effect of Maternal Fasting During Pregnancy WP-07-22 Douglas Almond and Bhashkar Mazumder

The Spending and Debt Response to Minimum Wage Hikes WP-07-23 Daniel Aaronson, Sumit Agarwal, and Eric French

The Impact of Mexican Immigrants on U.S. Wage Structure WP-07-24 Maude Toussaint-Comeau

A Leverage-based Model of Speculative Bubbles WP-08-01 Gadi Barlevy

Displacement, Asymmetric Information and Heterogeneous Human Capital WP-08-02 Luojia Hu and Christopher Taber

BankCaR (Bank Capital-at-Risk): A credit risk model for US commercial bank charge-offs WP-08-03 Jon Frye and Eduard Pelz

Bank Lending, Financing Constraints and SME Investment WP-08-04 Santiago Carbó-Valverde, Francisco Rodríguez-Fernández, and Gregory F. Udell

Global Inflation WP-08-05 Matteo Ciccarelli and Benoît Mojon

Scale and the Origins of Structural Change WP-08-06 Francisco J. Buera and Joseph P. Kaboski

Inventories, Lumpy Trade, and Large Devaluations WP-08-07 George Alessandria, Joseph P. Kaboski, and Virgiliu Midrigan 2 Working Paper Series (continued)

School Vouchers and Student Achievement: Recent Evidence, Remaining Questions WP-08-08 Cecilia Elena Rouse and Lisa Barrow

Does It Pay to Read Your Junk Mail? Evidence of the Effect of Advertising on Home Equity Credit Choices WP-08-09 Sumit Agarwal and Brent W. Ambrose

The Choice between Arm’s-Length and Relationship Debt: Evidence from eLoans WP-08-10 Sumit Agarwal and Robert Hauswald

Consumer Choice and Merchant Acceptance of Payment Media WP-08-11 Wilko Bolt and Sujit Chakravorti

Investment Shocks and Business Cycles WP-08-12 Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti

New Vehicle Characteristics and the Cost of the Corporate Average Fuel Economy Standard WP-08-13 Thomas Klier and Joshua Linn

Realized Volatility WP-08-14 Torben G. Andersen and Luca Benzoni

Revenue Bubbles and Structural Deficits: What’s a state to do? WP-08-15 Richard Mattoon and Leslie McGranahan

The role of lenders in the home price boom WP-08-16 Richard J. Rosen

Bank Crises and Investor Confidence WP-08-17 Una Okonkwo Osili and Anna Paulson

Life Expectancy and Old Age Savings WP-08-18 Mariacristina De Nardi, Eric French, and John Bailey Jones

Remittance Behavior among New U.S. Immigrants WP-08-19 Katherine Meckel

Birth Cohort and the Black-White Achievement Gap: The Roles of Access and Health Soon After Birth WP-08-20 Kenneth Y. Chay, Jonathan Guryan, and Bhashkar Mazumder

Public Investment and Budget Rules for State vs. Local Governments WP-08-21 Marco Bassetto

Why Has Home Ownership Fallen Among the Young? WP-09-01 Jonas D.M. Fisher and Martin Gervais

Why do the Elderly Save? The Role of Medical Expenses WP-09-02 Mariacristina De Nardi, Eric French, and John Bailey Jones

3 Working Paper Series (continued)

Using Stock Returns to Identify Government Spending Shocks WP-09-03 Jonas D.M. Fisher and Ryan Peters

Stochastic Volatility WP-09-04 Torben G. Andersen and Luca Benzoni

The Effect of Disability Insurance Receipt on Labor Supply WP-09-05 Eric French and Jae Song

CEO Overconfidence and Dividend Policy WP-09-06 Sanjay Deshmukh, Anand M. Goel, and Keith M. Howe

Do Financial Counseling Mandates Improve Mortgage Choice and Performance? WP-09-07 Evidence from a Legislative Experiment Sumit Agarwal,Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet, and Douglas D. Evanoff

Perverse Incentives at the Banks? Evidence from a Natural Experiment WP-09-08 Sumit Agarwal and Faye H. Wang

Pay for Percentile WP-09-09 Gadi Barlevy and Derek Neal

The Life and Times of Nicolas Dutot WP-09-10 François R. Velde

Regulating Two-Sided Markets: An Empirical Investigation WP-09-11 Santiago Carbó Valverde, Sujit Chakravorti, and Francisco Rodriguez Fernandez

The Case of the Undying Debt WP-09-12 François R. Velde

Paying for Performance: The Education Impacts of a Community College Scholarship Program for Low-income Adults WP-09-13 Lisa Barrow, Lashawn Richburg-Hayes, Cecilia Elena Rouse, and Thomas Brock

Establishments Dynamics, Vacancies and Unemployment: A Neoclassical Synthesis WP-09-14 Marcelo Veracierto

The Price of Gasoline and the Demand for Fuel Economy: Evidence from Monthly New Vehicles Sales Data WP-09-15 Thomas Klier and Joshua Linn

Estimation of a Transformation Model with Truncation, Interval Observation and Time-Varying Covariates WP-09-16 Bo E. Honoré and Luojia Hu

Self-Enforcing Trade Agreements: Evidence from Antidumping Policy WP-09-17 Chad P. Bown and Meredith A. Crowley

Too much right can make a wrong: Setting the stage for the financial crisis WP-09-18 Richard J. Rosen 4 Working Paper Series (continued)

Can Structural Small Open Economy Models Account for the Influence of Foreign Disturbances? WP-09-19 Alejandro Justiniano and Bruce Preston

Liquidity Constraints of the Middle Class WP-09-20 Jeffrey R. Campbell and Zvi Hercowitz

Monetary Policy and Uncertainty in an Empirical Small Open Economy Model WP-09-21 Alejandro Justiniano and Bruce Preston

Firm boundaries and buyer-supplier match in market transaction: IT system procurement of U.S. credit unions WP-09-22 Yukako Ono and Junichi Suzuki

Health and the Savings of Insured Versus Uninsured, Working-Age Households in the U.S. WP-09-23 Maude Toussaint-Comeau and Jonathan Hartley

The Economics of “Radiator Springs:” Industry Dynamics, Sunk Costs, and Spatial Demand Shifts WP-09-24 Jeffrey R. Campbell and Thomas N. Hubbard

On the Relationship between Mobility, Population Growth, and Capital Spending in the United States WP-09-25 Marco Bassetto and Leslie McGranahan

The Impact of Rosenwald Schools on Black Achievement WP-09-26 Daniel Aaronson and Bhashkar Mazumder

Comment on “Letting Different Views about Business Cycles Compete” WP-10-01 Jonas D.M. Fisher

Macroeconomic Implications of Agglomeration WP-10-02 Morris A. Davis, Jonas D.M. Fisher and Toni M. Whited

Accounting for non-annuitization WP-10-03 Svetlana Pashchenko

Robustness and Macroeconomic Policy WP-10-04 Gadi Barlevy

Benefits of Relationship Banking: Evidence from Consumer Credit Markets WP-10-05 Sumit Agarwal, Souphala Chomsisengphet, Chunlin Liu, and Nicholas S. Souleles

The Effect of Sales Tax Holidays on Household Consumption Patterns WP-10-06 Nathan Marwell and Leslie McGranahan

Gathering Insights on the Forest from the Trees: A New Metric for Financial Conditions WP-10-07 Scott Brave and R. Andrew Butters

Identification of Models of the Labor Market WP-10-08 Eric French and Christopher Taber

5 Working Paper Series (continued)

Public Pensions and Labor Supply Over the Life Cycle WP-10-09 Eric French and John Jones

Explaining Asset Pricing Puzzles Associated with the 1987 Market Crash WP-10-10 Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein

Does Prenatal Sex Selection Improve Girls’ Well‐Being? Evidence from India WP-10-11 Luojia Hu and Analía Schlosser

Mortgage Choices and Housing Speculation WP-10-12 Gadi Barlevy and Jonas D.M. Fisher

Did Adhering to the Gold Standard Reduce the Cost of Capital? WP-10-13 Ron Alquist and Benjamin Chabot

Introduction to the Macroeconomic Dynamics: Special issues on money, credit, and liquidity WP-10-14 Ed Nosal, Christopher Waller, and Randall Wright

Summer Workshop on Money, Banking, Payments and Finance: An Overview WP-10-15 Ed Nosal and Randall Wright

Cognitive Abilities and Household Financial Decision Making WP-10-16 Sumit Agarwal and Bhashkar Mazumder

Complex Mortgages WP-10-17 Gene Amromin, Jennifer Huang, Clemens Sialm, and Edward Zhong

The Role of Housing in Labor Reallocation WP-10-18 Morris Davis, Jonas Fisher, and Marcelo Veracierto

Why Do Banks Reward their Customers to Use their Credit Cards? WP-10-19 Sumit Agarwal, Sujit Chakravorti, and Anna Lunn

The impact of the originate-to-distribute model on banks before and during the financial crisis WP-10-20 Richard J. Rosen

Simple Markov-Perfect Industry Dynamics WP-10-21 Jaap H. Abbring, Jeffrey R. Campbell, and Nan Yang